Sr Examen
Integral de sin^3x/(cosx-cos^2x) dx
Solución
Ha introducido
[src]
pi -- 3 / | | 3 | sin (x) | ---------------- dx | 2 | cos(x) - cos (x) | / pi -- 6
$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin^{3}{\left(x \right)}}{- \cos^{2}{\left(x \right)} + \cos{\left(x \right)}}\, dx$$
Integral(sin(x)^3/(cos(x) - cos(x)^2), (x, pi/6, pi/3))
Respuesta (Indefinida)
[src]
/ | / 2/x\\ / /x\\ / /x\\ 2/x\ / 2/x\\ 2/x\ / /x\\ 2/x\ / /x\\ | 3 log|1 + tan |-|| log|1 + tan|-|| log|-1 + tan|-|| tan |-|*log|1 + tan |-|| tan |-|*log|1 + tan|-|| tan |-|*log|-1 + tan|-|| | sin (x) 2 \ \2// \ \2// \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// | ---------------- dx = C - ----------- + ---------------- - --------------- - ---------------- + ------------------------ - ----------------------- - ------------------------ | 2 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | cos(x) - cos (x) 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| | \2/ \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{\sin^{3}{\left(x \right)}}{- \cos^{2}{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
Gráfica
Respuesta
[src]
/ 2\ 2 2 2 / 2\
/ ___\ / ___\ / ___\ / ___\ | / ___\ | / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ |
3 | \/ 3 | | \/ 3 | 2 pi*I + log\-1 + \/ 3 / log\3 - \/ 3 / log\1 + \2 - \/ 3 / / \2 - \/ 3 / *\pi*I + log\-1 + \/ 3 // \2 - \/ 3 / *log\3 - \/ 3 / \2 - \/ 3 / *log\1 + \2 - \/ 3 / /
- - - log|1 - -----| - log|1 + -----| + ---------------- + ---------------------- + ---------------- - pi*I - --------------------- + ------------------------------------- + --------------------------- - ---------------------------------- + log(4/3)
2 \ 3 / \ 3 / 2 2 2 2 2 2 2
/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\
1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 /
$$- \frac{3}{2} - \log{\left(\frac{\sqrt{3}}{3} + 1 \right)} - \frac{\log{\left(\left(2 - \sqrt{3}\right)^{2} + 1 \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} - \frac{\left(2 - \sqrt{3}\right)^{2} \log{\left(\left(2 - \sqrt{3}\right)^{2} + 1 \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\left(2 - \sqrt{3}\right)^{2} \log{\left(3 - \sqrt{3} \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\log{\left(3 - \sqrt{3} \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \log{\left(\frac{4}{3} \right)} - \log{\left(1 - \frac{\sqrt{3}}{3} \right)} + \frac{2}{\left(2 - \sqrt{3}\right)^{2} + 1} - i \pi + \frac{\left(2 - \sqrt{3}\right)^{2} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\log{\left(-1 + \sqrt{3} \right)} + i \pi}{\left(2 - \sqrt{3}\right)^{2} + 1}$$
=
=
/ 2\ 2 2 2 / 2\
/ ___\ / ___\ / ___\ / ___\ | / ___\ | / ___\ / / ___\\ / ___\ / ___\ / ___\ | / ___\ |
3 | \/ 3 | | \/ 3 | 2 pi*I + log\-1 + \/ 3 / log\3 - \/ 3 / log\1 + \2 - \/ 3 / / \2 - \/ 3 / *\pi*I + log\-1 + \/ 3 // \2 - \/ 3 / *log\3 - \/ 3 / \2 - \/ 3 / *log\1 + \2 - \/ 3 / /
- - - log|1 - -----| - log|1 + -----| + ---------------- + ---------------------- + ---------------- - pi*I - --------------------- + ------------------------------------- + --------------------------- - ---------------------------------- + log(4/3)
2 \ 3 / \ 3 / 2 2 2 2 2 2 2
/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\
1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 / 1 + \2 - \/ 3 /
$$- \frac{3}{2} - \log{\left(\frac{\sqrt{3}}{3} + 1 \right)} - \frac{\log{\left(\left(2 - \sqrt{3}\right)^{2} + 1 \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} - \frac{\left(2 - \sqrt{3}\right)^{2} \log{\left(\left(2 - \sqrt{3}\right)^{2} + 1 \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\left(2 - \sqrt{3}\right)^{2} \log{\left(3 - \sqrt{3} \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\log{\left(3 - \sqrt{3} \right)}}{\left(2 - \sqrt{3}\right)^{2} + 1} + \log{\left(\frac{4}{3} \right)} - \log{\left(1 - \frac{\sqrt{3}}{3} \right)} + \frac{2}{\left(2 - \sqrt{3}\right)^{2} + 1} - i \pi + \frac{\left(2 - \sqrt{3}\right)^{2} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{\left(2 - \sqrt{3}\right)^{2} + 1} + \frac{\log{\left(-1 + \sqrt{3} \right)} + i \pi}{\left(2 - \sqrt{3}\right)^{2} + 1}$$
-3/2 - log(1 - sqrt(3)/3) - log(1 + sqrt(3)/3) + 2/(1 + (2 - sqrt(3))^2) + (pi*i + log(-1 + sqrt(3)))/(1 + (2 - sqrt(3))^2) + log(3 - sqrt(3))/(1 + (2 - sqrt(3))^2) - pi*i - log(1 + (2 - sqrt(3))^2)/(1 + (2 - sqrt(3))^2) + (2 - sqrt(3))^2*(pi*i + log(-1 + sqrt(3)))/(1 + (2 - sqrt(3))^2) + (2 - sqrt(3))^2*log(3 - sqrt(3))/(1 + (2 - sqrt(3))^2) - (2 - sqrt(3))^2*log(1 + (2 - sqrt(3))^2)/(1 + (2 - sqrt(3))^2) + log(4/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.