Sr Examen
Integral de √(2-cbrt(x))/cbrt(x) dx
Solución
Ha introducido
[src]
1 / | | ___________ | / 3 ___ | \/ 2 - \/ x | -------------- dx | 3 ___ | \/ x | / 0
$$\int\limits_{0}^{1} \frac{\sqrt{2 - \sqrt[3]{x}}}{\sqrt[3]{x}}\, dx$$
Integral(sqrt(2 - x^(1/3))/x^(1/3), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ | | ___________ 5/2 | / 3 ___ 3/2 / 3 ___\ | \/ 2 - \/ x / 3 ___\ 6*\2 - \/ x / | -------------- dx = C - 4*\2 - \/ x / + ---------------- | 3 ___ 5 | \/ x | /
$$\int \frac{\sqrt{2 - \sqrt[3]{x}}}{\sqrt[3]{x}}\, dx = C + \frac{6 \left(2 - \sqrt[3]{x}\right)^{\frac{5}{2}}}{5} - 4 \left(2 - \sqrt[3]{x}\right)^{\frac{3}{2}}$$
Gráfica
Respuesta
[src]
1 / | | / / ___ 2/3 ___ 3 ___\ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ | | 4/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | 5/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | ____________ ____________ ____________ ____________ ___ 2 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 5/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 7/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 4/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | | | 3 ___ 2/3 64*x *|- ------------- + --------------| 32*x *|- ------------- + --------------| ___ 5/3 ___ / 3 ___ ___ 4/3 / 3 ___ ___ 2/3 / 3 ___ ___ 4/3 ___ ___ 2/3 ___ 3 ___ / 3 ___ 16*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 8*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 6*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 32*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 3 ___ | | 256*\/ x 160*x \ 3 3 / \ 3 3 / I*\/ 2 *x 32*I*x*\/ 2 *\/ -2 + \/ x 14*I*\/ 2 *x *\/ -2 + \/ x 40*I*\/ 2 *x *\/ -2 + \/ x 8*I*\/ 2 *x 4*I*x*\/ 2 16*I*\/ 2 *x 128*I*\/ 2 *\/ x *\/ -2 + \/ x \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / \/ x | |- ---------------------------------- + ---------------------------------- - ------------------------------------------ + ------------------------------------------ + ------------------------------------------------ - ------------------------------ + ------------------------------- - ---------------------------------- - -------------------------------------------------- - -------------------------------------------------- + -------------------------------------------------- + ---------------------------------- - ---------------------------------------------------------------- - ----------------------------------------------------------------- + ----------------------------------------------------------------- + ------------------------------------------------------------------ for ----- > 1 | | / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ 2 2 ____________ ___ 4/3 ___ 5/3 ___ 4/3 ___ 5/3 / ___ 4/3 ___ 5/3\ ____________ ____________ ____________ / ___ 4/3 ___ 5/3\ 2 2 2 2 2 | | 3*\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ - 10*\/ 2 *x + 5*\/ 2 *x - 10*\/ 2 *x + 5*\/ 2 *x 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ | | \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / | < dx | | / ___ 2/3 ___ 3 ___\ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ | | 4/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | 5/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | ___________ ___________ ___________ ___________ ___ 2 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 5/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 7/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 4/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | | | 3 ___ 2/3 64*x *|- ------------- + --------------| 32*x *|- ------------- + --------------| ___ 5/3 ___ / 3 ___ ___ 4/3 / 3 ___ ___ 2/3 / 3 ___ ___ 2/3 ___ ___ 4/3 ___ 3 ___ / 3 ___ 16*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 8*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 6*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 32*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| | | 256*\/ x 160*x \ 3 3 / \ 3 3 / \/ 2 *x 32*x*\/ 2 *\/ 2 - \/ x 14*\/ 2 *x *\/ 2 - \/ x 40*\/ 2 *x *\/ 2 - \/ x 16*\/ 2 *x 4*x*\/ 2 8*\/ 2 *x 128*\/ 2 *\/ x *\/ 2 - \/ x \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / | | - ---------------------------------- + ---------------------------------- - ------------------------------------------ + ------------------------------------------ - ----------------------------------------------- - ------------------------------ + ------------------------------ - ---------------------------------- - ------------------------------------------------- + ------------------------------------------------- + ------------------------------------------------- + ---------------------------------- - ------------------------------------------------------------- - -------------------------------------------------------------- + -------------------------------------------------------------- + --------------------------------------------------------------- otherwise | | / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ 2 2 ___________ ___ 4/3 ___ 5/3 ___ 4/3 ___ 5/3 / ___ 4/3 ___ 5/3\ ___________ ___________ ___________ / ___ 4/3 ___ 5/3\ 2 2 2 2 | | 3*\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ - 10*\/ 2 *x + 5*\/ 2 *x - 10*\/ 2 *x + 5*\/ 2 *x 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ | \ \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{6 \sqrt{2} i x^{\frac{7}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} i x^{\frac{5}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{32 x^{\frac{5}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{\sqrt{2} i x^{\frac{5}{3}}}{\sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{32 \sqrt{2} i x^{\frac{4}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{14 \sqrt{2} i x^{\frac{4}{3}} \sqrt{\sqrt[3]{x} - 2}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{64 x^{\frac{4}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} i x^{\frac{4}{3}}}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{40 \sqrt{2} i x^{\frac{2}{3}} \sqrt{\sqrt[3]{x} - 2}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{160 x^{\frac{2}{3}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{16 \sqrt{2} i x^{\frac{2}{3}}}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{128 \sqrt{2} i \sqrt[3]{x} \sqrt{\sqrt[3]{x} - 2}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{256 \sqrt[3]{x}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} i x^{2} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{32 \sqrt{2} i x \sqrt{\sqrt[3]{x} - 2}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{4 \sqrt{2} i x}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} & \text{for}\: \frac{\sqrt[3]{x}}{2} > 1 \\\frac{6 \sqrt{2} x^{\frac{7}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} x^{\frac{5}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{32 x^{\frac{5}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{\sqrt{2} x^{\frac{5}{3}}}{\sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{32 \sqrt{2} x^{\frac{4}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{14 \sqrt{2} x^{\frac{4}{3}} \sqrt{2 - \sqrt[3]{x}}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{64 x^{\frac{4}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{8 \sqrt{2} x^{\frac{4}{3}}}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{40 \sqrt{2} x^{\frac{2}{3}} \sqrt{2 - \sqrt[3]{x}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{160 x^{\frac{2}{3}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} x^{\frac{2}{3}}}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{128 \sqrt{2} \sqrt[3]{x} \sqrt{2 - \sqrt[3]{x}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{256 \sqrt[3]{x}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} x^{2} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{32 \sqrt{2} x \sqrt{2 - \sqrt[3]{x}}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} + \frac{4 \sqrt{2} x}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / / ___ 2/3 ___ 3 ___\ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ ____________ / ___ 2/3 ___ 3 ___\ | | 4/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | 5/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | ____________ ____________ ____________ ____________ ___ 2 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 5/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 7/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 4/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | | | 3 ___ 2/3 64*x *|- ------------- + --------------| 32*x *|- ------------- + --------------| ___ 5/3 ___ / 3 ___ ___ 4/3 / 3 ___ ___ 2/3 / 3 ___ ___ 4/3 ___ ___ 2/3 ___ 3 ___ / 3 ___ 16*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 8*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 6*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 32*I*\/ 2 *x *\/ -2 + \/ x *|- ------------- + --------------| 3 ___ | | 256*\/ x 160*x \ 3 3 / \ 3 3 / I*\/ 2 *x 32*I*x*\/ 2 *\/ -2 + \/ x 14*I*\/ 2 *x *\/ -2 + \/ x 40*I*\/ 2 *x *\/ -2 + \/ x 8*I*\/ 2 *x 4*I*x*\/ 2 16*I*\/ 2 *x 128*I*\/ 2 *\/ x *\/ -2 + \/ x \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / \/ x | |- ---------------------------------- + ---------------------------------- - ------------------------------------------ + ------------------------------------------ + ------------------------------------------------ - ------------------------------ + ------------------------------- - ---------------------------------- - -------------------------------------------------- - -------------------------------------------------- + -------------------------------------------------- + ---------------------------------- - ---------------------------------------------------------------- - ----------------------------------------------------------------- + ----------------------------------------------------------------- + ------------------------------------------------------------------ for ----- > 1 | | / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ 2 2 ____________ ___ 4/3 ___ 5/3 ___ 4/3 ___ 5/3 / ___ 4/3 ___ 5/3\ ____________ ____________ ____________ / ___ 4/3 ___ 5/3\ 2 2 2 2 2 | | 3*\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ - 10*\/ 2 *x + 5*\/ 2 *x - 10*\/ 2 *x + 5*\/ 2 *x 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ | | \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ -2 + \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / | < dx | | / ___ 2/3 ___ 3 ___\ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ ___________ / ___ 2/3 ___ 3 ___\ | | 4/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | 5/3 | 25*\/ 2 *x 40*\/ 2 *\/ x | ___________ ___________ ___________ ___________ ___ 2 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 5/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 7/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | ___ 4/3 / 3 ___ | 25*\/ 2 *x 40*\/ 2 *\/ x | | | 3 ___ 2/3 64*x *|- ------------- + --------------| 32*x *|- ------------- + --------------| ___ 5/3 ___ / 3 ___ ___ 4/3 / 3 ___ ___ 2/3 / 3 ___ ___ 2/3 ___ ___ 4/3 ___ 3 ___ / 3 ___ 16*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 8*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 6*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| 32*\/ 2 *x *\/ 2 - \/ x *|- ------------- + --------------| | | 256*\/ x 160*x \ 3 3 / \ 3 3 / \/ 2 *x 32*x*\/ 2 *\/ 2 - \/ x 14*\/ 2 *x *\/ 2 - \/ x 40*\/ 2 *x *\/ 2 - \/ x 16*\/ 2 *x 4*x*\/ 2 8*\/ 2 *x 128*\/ 2 *\/ x *\/ 2 - \/ x \ 3 3 / \ 3 3 / \ 3 3 / \ 3 3 / | | - ---------------------------------- + ---------------------------------- - ------------------------------------------ + ------------------------------------------ - ----------------------------------------------- - ------------------------------ + ------------------------------ - ---------------------------------- - ------------------------------------------------- + ------------------------------------------------- + ------------------------------------------------- + ---------------------------------- - ------------------------------------------------------------- - -------------------------------------------------------------- + -------------------------------------------------------------- + --------------------------------------------------------------- otherwise | | / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ 2 2 ___________ ___ 4/3 ___ 5/3 ___ 4/3 ___ 5/3 / ___ 4/3 ___ 5/3\ ___________ ___________ ___________ / ___ 4/3 ___ 5/3\ 2 2 2 2 | | 3*\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ - 10*\/ 2 *x + 5*\/ 2 *x - 10*\/ 2 *x + 5*\/ 2 *x 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ / 3 ___ / ___ 4/3 ___ 5/3\ 3*\- 10*\/ 2 *x + 5*\/ 2 *x / / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ / ___ 4/3 ___ 5/3\ | \ \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / 3*\/ 2 - \/ x *\- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / \- 10*\/ 2 *x + 5*\/ 2 *x / | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{6 \sqrt{2} i x^{\frac{7}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} i x^{\frac{5}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{32 x^{\frac{5}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{\sqrt{2} i x^{\frac{5}{3}}}{\sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{32 \sqrt{2} i x^{\frac{4}{3}} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{14 \sqrt{2} i x^{\frac{4}{3}} \sqrt{\sqrt[3]{x} - 2}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{64 x^{\frac{4}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} i x^{\frac{4}{3}}}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{40 \sqrt{2} i x^{\frac{2}{3}} \sqrt{\sqrt[3]{x} - 2}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{160 x^{\frac{2}{3}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{16 \sqrt{2} i x^{\frac{2}{3}}}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{128 \sqrt{2} i \sqrt[3]{x} \sqrt{\sqrt[3]{x} - 2}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{256 \sqrt[3]{x}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} i x^{2} \sqrt{\sqrt[3]{x} - 2} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{32 \sqrt{2} i x \sqrt{\sqrt[3]{x} - 2}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{4 \sqrt{2} i x}{3 \sqrt{\sqrt[3]{x} - 2} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} & \text{for}\: \frac{\sqrt[3]{x}}{2} > 1 \\\frac{6 \sqrt{2} x^{\frac{7}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{8 \sqrt{2} x^{\frac{5}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{32 x^{\frac{5}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{\sqrt{2} x^{\frac{5}{3}}}{\sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{32 \sqrt{2} x^{\frac{4}{3}} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{14 \sqrt{2} x^{\frac{4}{3}} \sqrt{2 - \sqrt[3]{x}}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} - \frac{64 x^{\frac{4}{3}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} + \frac{8 \sqrt{2} x^{\frac{4}{3}}}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{40 \sqrt{2} x^{\frac{2}{3}} \sqrt{2 - \sqrt[3]{x}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{160 x^{\frac{2}{3}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} x^{\frac{2}{3}}}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} + \frac{128 \sqrt{2} \sqrt[3]{x} \sqrt{2 - \sqrt[3]{x}}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{256 \sqrt[3]{x}}{3 \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} - \frac{16 \sqrt{2} x^{2} \sqrt{2 - \sqrt[3]{x}} \left(- \frac{25 \sqrt{2} x^{\frac{2}{3}}}{3} + \frac{40 \sqrt{2} \sqrt[3]{x}}{3}\right)}{\left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)^{2}} - \frac{32 \sqrt{2} x \sqrt{2 - \sqrt[3]{x}}}{5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}} + \frac{4 \sqrt{2} x}{3 \sqrt{2 - \sqrt[3]{x}} \left(5 \sqrt{2} x^{\frac{5}{3}} - 10 \sqrt{2} x^{\frac{4}{3}}\right)} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-256*x^(1/3)/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 160*x^(2/3)/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 64*x^(4/3)*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 32*x^(5/3)*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + i*sqrt(2)*x^(5/3)/(sqrt(-2 + x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 32*i*x*sqrt(2)*sqrt(-2 + x^(1/3))/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3)) + 14*i*sqrt(2)*x^(4/3)*sqrt(-2 + x^(1/3))/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3)) - 40*i*sqrt(2)*x^(2/3)*sqrt(-2 + x^(1/3))/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 8*i*sqrt(2)*x^(4/3)/(3*sqrt(-2 + x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 4*i*x*sqrt(2)/(3*sqrt(-2 + x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 16*i*sqrt(2)*x^(2/3)/(3*sqrt(-2 + x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 128*i*sqrt(2)*x^(1/3)*sqrt(-2 + x^(1/3))/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 16*i*sqrt(2)*x^2*sqrt(-2 + x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 - 8*i*sqrt(2)*x^(5/3)*sqrt(-2 + x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 6*i*sqrt(2)*x^(7/3)*sqrt(-2 + x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 32*i*sqrt(2)*x^(4/3)*sqrt(-2 + x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2, x^(1/3)/2 > 1), (-256*x^(1/3)/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 160*x^(2/3)/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 64*x^(4/3)*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 32*x^(5/3)*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 - sqrt(2)*x^(5/3)/(sqrt(2 - x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 32*x*sqrt(2)*sqrt(2 - x^(1/3))/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3)) + 14*sqrt(2)*x^(4/3)*sqrt(2 - x^(1/3))/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3)) - 40*sqrt(2)*x^(2/3)*sqrt(2 - x^(1/3))/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 16*sqrt(2)*x^(2/3)/(3*sqrt(2 - x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 4*x*sqrt(2)/(3*sqrt(2 - x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 8*sqrt(2)*x^(4/3)/(3*sqrt(2 - x^(1/3))*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) + 128*sqrt(2)*x^(1/3)*sqrt(2 - x^(1/3))/(3*(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))) - 16*sqrt(2)*x^2*sqrt(2 - x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 - 8*sqrt(2)*x^(5/3)*sqrt(2 - x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 6*sqrt(2)*x^(7/3)*sqrt(2 - x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2 + 32*sqrt(2)*x^(4/3)*sqrt(2 - x^(1/3))*(-25*sqrt(2)*x^(2/3)/3 + 40*sqrt(2)*x^(1/3)/3)/(-10*sqrt(2)*x^(4/3) + 5*sqrt(2)*x^(5/3))^2, True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.