Sr Examen
Integral de xsinx/cos^3x dx
Solución
Ha introducido
[src]
pi -- 4 / | | x*sin(x) | -------- dx | 3 | cos (x) | / 0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx$$
Integral((x*sin(x))/cos(x)^3, (x, 0, pi/4))
Respuesta (Indefinida)
[src]
/ /x\ 3/x\ 4/x\ 2/x\ | 2*tan|-| 2*tan |-| x*tan |-| 2*x*tan |-| | x*sin(x) x \2/ \2/ \2/ \2/ | -------- dx = C + ------------------------- - ------------------------- + ------------------------- + ------------------------- + ------------------------- | 3 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ | cos (x) 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| | \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx = C + \frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$
Gráfica
Respuesta
[src]
3 2 4
/ ___\ / ___\ / ___\ / ___\
2*\-1 + \/ 2 / 2*\-1 + \/ 2 / pi pi*\-1 + \/ 2 / pi*\-1 + \/ 2 /
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
2 4 2 4 / 2 4\ / 2 4\ / 2 4\
/ ___\ / ___\ / ___\ / ___\ | / ___\ / ___\ | | / ___\ / ___\ | | / ___\ / ___\ |
2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / 2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / 4*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / / 2*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / / 4*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / /
$$- \frac{2 \left(-1 + \sqrt{2}\right)}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{4}}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{2 \left(-1 + \sqrt{2}\right)^{3}}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{2}}{2 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{\pi}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)}$$
=
=
3 2 4
/ ___\ / ___\ / ___\ / ___\
2*\-1 + \/ 2 / 2*\-1 + \/ 2 / pi pi*\-1 + \/ 2 / pi*\-1 + \/ 2 /
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
2 4 2 4 / 2 4\ / 2 4\ / 2 4\
/ ___\ / ___\ / ___\ / ___\ | / ___\ / ___\ | | / ___\ / ___\ | | / ___\ / ___\ |
2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / 2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / 4*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / / 2*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / / 4*\2 - 4*\-1 + \/ 2 / + 2*\-1 + \/ 2 / /
$$- \frac{2 \left(-1 + \sqrt{2}\right)}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{4}}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{2 \left(-1 + \sqrt{2}\right)^{3}}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{2}}{2 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{\pi}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)}$$
-2*(-1 + sqrt(2))/(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4) + 2*(-1 + sqrt(2))^3/(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4) + pi/(4*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4)) + pi*(-1 + sqrt(2))^2/(2*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4)) + pi*(-1 + sqrt(2))^4/(4*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.