Sr Examen
Integral de x*(1-x^4)^1/2 dx
Solución
Ha introducido
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1 / | | ________ | / 4 | x*\/ 1 - x dx | / 0
$$\int\limits_{0}^{1} x \sqrt{1 - x^{4}}\, dx$$
Integral(x*sqrt(1 - x^4), (x, 0, 1))
Respuesta (Indefinida)
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// / 2\ 2 6 \
|| I*acosh\x / I*x I*x | 4| |
/ ||- ----------- - -------------- + -------------- for |x | > 1|
| || 4 _________ _________ |
| ________ || / 4 / 4 |
| / 4 || 4*\/ -1 + x 4*\/ -1 + x |
| x*\/ 1 - x dx = C + |< |
| || ________ |
/ || / 2\ 2 / 4 |
|| asin\x / x *\/ 1 - x |
|| -------- + -------------- otherwise |
|| 4 4 |
\\ /
$$\int x \sqrt{1 - x^{4}}\, dx = C + \begin{cases} \frac{i x^{6}}{4 \sqrt{x^{4} - 1}} - \frac{i x^{2}}{4 \sqrt{x^{4} - 1}} - \frac{i \operatorname{acosh}{\left(x^{2} \right)}}{4} & \text{for}\: \left|{x^{4}}\right| > 1 \\\frac{x^{2} \sqrt{1 - x^{4}}}{4} + \frac{\operatorname{asin}{\left(x^{2} \right)}}{4} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 5 9 5 | | I*x I*x I*x 3*I*x 4 | |-------------- - ------------ - -------------- + -------------- for x > 1 | | 3/2 _________ 3/2 _________ | | / 4\ / 4 / 4\ / 4 | |2*\-1 + x / \/ -1 + x 2*\-1 + x / 2*\/ -1 + x | | | < ________ dx | | / 4 5 | | x*\/ 1 - x x x | | ------------- + ------------- - ------------- otherwise | | 2 ________ ________ | | / 4 / 4 | | 2*\/ 1 - x 2*\/ 1 - x | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{9}}{2 \left(x^{4} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{5}}{2 \sqrt{x^{4} - 1}} + \frac{i x^{5}}{2 \left(x^{4} - 1\right)^{\frac{3}{2}}} - \frac{i x}{\sqrt{x^{4} - 1}} & \text{for}\: x^{4} > 1 \\- \frac{x^{5}}{2 \sqrt{1 - x^{4}}} + \frac{x \sqrt{1 - x^{4}}}{2} + \frac{x}{2 \sqrt{1 - x^{4}}} & \text{otherwise} \end{cases}\, dx$$
=
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1 / | | / 5 9 5 | | I*x I*x I*x 3*I*x 4 | |-------------- - ------------ - -------------- + -------------- for x > 1 | | 3/2 _________ 3/2 _________ | | / 4\ / 4 / 4\ / 4 | |2*\-1 + x / \/ -1 + x 2*\-1 + x / 2*\/ -1 + x | | | < ________ dx | | / 4 5 | | x*\/ 1 - x x x | | ------------- + ------------- - ------------- otherwise | | 2 ________ ________ | | / 4 / 4 | | 2*\/ 1 - x 2*\/ 1 - x | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{9}}{2 \left(x^{4} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{5}}{2 \sqrt{x^{4} - 1}} + \frac{i x^{5}}{2 \left(x^{4} - 1\right)^{\frac{3}{2}}} - \frac{i x}{\sqrt{x^{4} - 1}} & \text{for}\: x^{4} > 1 \\- \frac{x^{5}}{2 \sqrt{1 - x^{4}}} + \frac{x \sqrt{1 - x^{4}}}{2} + \frac{x}{2 \sqrt{1 - x^{4}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((i*x^5/(2*(-1 + x^4)^(3/2)) - i*x/sqrt(-1 + x^4) - i*x^9/(2*(-1 + x^4)^(3/2)) + 3*i*x^5/(2*sqrt(-1 + x^4)), x^4 > 1), (x*sqrt(1 - x^4)/2 + x/(2*sqrt(1 - x^4)) - x^5/(2*sqrt(1 - x^4)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.