Sr Examen
Integral de (-2+2x(0.8-2x)/(x^2-1)+12(0.8x-x^2)/sqrt(1-x^2))(0.8x^4-x^5) dx
Solución
Ha introducido
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4/5 / | | / /4*x 2\\ | | 12*|--- - x || / 4 \ | | 2*x*(4/5 - 2*x) \ 5 /| |4*x 5| | |-2 + --------------- + -------------|*|---- - x | dx | | 2 ________ | \ 5 / | | x - 1 / 2 | | \ \/ 1 - x / | / 0
$$\int\limits_{0}^{\frac{4}{5}} \left(- x^{5} + \frac{4 x^{4}}{5}\right) \left(\left(\frac{2 x \left(\frac{4}{5} - 2 x\right)}{x^{2} - 1} - 2\right) + \frac{12 \left(- x^{2} + \frac{4 x}{5}\right)}{\sqrt{1 - x^{2}}}\right)\, dx$$
Integral((-2 + ((2*x)*(4/5 - 2*x))/(x^2 - 1) + (12*(4*x/5 - x^2))/sqrt(1 - x^2))*(4*x^4/5 - x^5), (x, 0, 4/5))
Respuesta (Indefinida)
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/ // ________ 3/2 ________ \ // 5/2 3/2 \ | || / 2 3 / 2\ / 2 / 2\ | || ________ / 2\ / 2\ | | / /4*x 2\\ 96*|<5*asin(x) x*\/ 1 - x x *\1 - x / 3*x*\/ 1 - x *\1 - 2*x / | 192*|< / 2 \1 - x / 2*\1 - x / | | | 12*|--- - x || / 4 \ // 5/2 7/2 \ ||--------- - ------------- + -------------- + -------------------------- for And(x > -1, x < 1)| 5 3 4 2 / 2\ ||- \/ 1 - x - ----------- + ------------- for And(x > -1, x < 1)| | | 2*x*(4/5 - 2*x) \ 5 /| |4*x 5| 6 || 3/2 ________ / 2\ / 2\ | \\ 16 2 6 16 / 32*x 24*x 12*log(-1 + x) 8*x 12*log(1 + x) 33*x 66*x 66*log\-1 + x / \\ 5 3 / | |-2 + --------------- + -------------|*|---- - x | dx = C + x + 12*| -1, x < 1)| 5 25 5 5 5 5 25 25 25 25 | | x - 1 / 2 | \\ 5 7 / | \ \/ 1 - x / | /
$$\int \left(- x^{5} + \frac{4 x^{4}}{5}\right) \left(\left(\frac{2 x \left(\frac{4}{5} - 2 x\right)}{x^{2} - 1} - 2\right) + \frac{12 \left(- x^{2} + \frac{4 x}{5}\right)}{\sqrt{1 - x^{2}}}\right)\, dx = C + x^{6} - \frac{32 x^{5}}{25} + \frac{33 x^{4}}{25} - \frac{8 x^{3}}{5} + \frac{66 x^{2}}{25} - \frac{24 x}{5} + \frac{192 \left(\begin{cases} - \frac{\left(1 - x^{2}\right)^{\frac{5}{2}}}{5} + \frac{2 \left(1 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{1 - x^{2}} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right)}{25} - \frac{96 \left(\begin{cases} \frac{x^{3} \left(1 - x^{2}\right)^{\frac{3}{2}}}{6} + \frac{3 x \left(1 - 2 x^{2}\right) \sqrt{1 - x^{2}}}{16} - \frac{x \sqrt{1 - x^{2}}}{2} + \frac{5 \operatorname{asin}{\left(x \right)}}{16} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right)}{5} + 12 \left(\begin{cases} \frac{\left(1 - x^{2}\right)^{\frac{7}{2}}}{7} - \frac{3 \left(1 - x^{2}\right)^{\frac{5}{2}}}{5} + \left(1 - x^{2}\right)^{\frac{3}{2}} - \sqrt{1 - x^{2}} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right) - \frac{12 \log{\left(x - 1 \right)}}{5} + \frac{12 \log{\left(x + 1 \right)}}{5} + \frac{66 \log{\left(x^{2} - 1 \right)}}{25}$$
Respuesta
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8181368 6*log(5) 126*log(9/5) ------- - 6*asin(4/5) - -------- + ------------ 2734375 25 25
$$- 6 \operatorname{asin}{\left(\frac{4}{5} \right)} - \frac{6 \log{\left(5 \right)}}{25} + \frac{126 \log{\left(\frac{9}{5} \right)}}{25} + \frac{8181368}{2734375}$$
=
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8181368 6*log(5) 126*log(9/5) ------- - 6*asin(4/5) - -------- + ------------ 2734375 25 25
$$- 6 \operatorname{asin}{\left(\frac{4}{5} \right)} - \frac{6 \log{\left(5 \right)}}{25} + \frac{126 \log{\left(\frac{9}{5} \right)}}{25} + \frac{8181368}{2734375}$$
8181368/2734375 - 6*asin(4/5) - 6*log(5)/25 + 126*log(9/5)/25
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.