Sr Examen
Integral de cos^3(x)*e^(x) dx
Solución
Ha introducido
[src]
5*pi ---- 12 / | | 3 x | cos (x)*E dx | / pi -- 3
$$\int\limits_{\frac{\pi}{3}}^{\frac{5 \pi}{12}} e^{x} \cos^{3}{\left(x \right)}\, dx$$
Integral(cos(x)^3*E^x, (x, pi/3, 5*pi/12))
Respuesta (Indefinida)
[src]
/ | 3 x 3 x 2 x 2 x | 3 x 2*cos (x)*e 3*sin (x)*e 3*cos (x)*e *sin(x) 3*sin (x)*cos(x)*e | cos (x)*E dx = C + ------------ + ------------ + ------------------- + ------------------- | 5 10 5 10 /
$$\int e^{x} \cos^{3}{\left(x \right)}\, dx = C + \frac{3 e^{x} \sin^{3}{\left(x \right)}}{10} + \frac{3 e^{x} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{10} + \frac{3 e^{x} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{5} + \frac{2 e^{x} \cos^{3}{\left(x \right)}}{5}$$
Gráfica
Respuesta
[src]
3 5*pi 3 5*pi 2 5*pi 2 5*pi
pi pi / ___ ___\ ---- / ___ ___\ ---- / ___ ___\ / ___ ___\ ---- / ___ ___\ / ___ ___\ ----
-- -- | \/ 2 \/ 6 | 12 |\/ 2 \/ 6 | 12 | \/ 2 \/ 6 | |\/ 2 \/ 6 | 12 |\/ 2 \/ 6 | | \/ 2 \/ 6 | 12
3 ___ 3 2*|- ----- + -----| *e 3*|----- + -----| *e 3*|- ----- + -----| *|----- + -----|*e 3*|----- + -----| *|- ----- + -----|*e
13*e 3*\/ 3 *e \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 /
- ------ - ----------- + -------------------------- + ------------------------ + ------------------------------------------ + ------------------------------------------
80 16 5 10 5 10
$$- \frac{3 \sqrt{3} e^{\frac{\pi}{3}}}{16} - \frac{13 e^{\frac{\pi}{3}}}{80} + \frac{2 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} e^{\frac{5 \pi}{12}}}{5} + \frac{3 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) e^{\frac{5 \pi}{12}}}{5} + \frac{3 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} e^{\frac{5 \pi}{12}}}{10} + \frac{3 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} e^{\frac{5 \pi}{12}}}{10}$$
=
=
3 5*pi 3 5*pi 2 5*pi 2 5*pi
pi pi / ___ ___\ ---- / ___ ___\ ---- / ___ ___\ / ___ ___\ ---- / ___ ___\ / ___ ___\ ----
-- -- | \/ 2 \/ 6 | 12 |\/ 2 \/ 6 | 12 | \/ 2 \/ 6 | |\/ 2 \/ 6 | 12 |\/ 2 \/ 6 | | \/ 2 \/ 6 | 12
3 ___ 3 2*|- ----- + -----| *e 3*|----- + -----| *e 3*|- ----- + -----| *|----- + -----|*e 3*|----- + -----| *|- ----- + -----|*e
13*e 3*\/ 3 *e \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 /
- ------ - ----------- + -------------------------- + ------------------------ + ------------------------------------------ + ------------------------------------------
80 16 5 10 5 10
$$- \frac{3 \sqrt{3} e^{\frac{\pi}{3}}}{16} - \frac{13 e^{\frac{\pi}{3}}}{80} + \frac{2 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} e^{\frac{5 \pi}{12}}}{5} + \frac{3 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) e^{\frac{5 \pi}{12}}}{5} + \frac{3 \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} e^{\frac{5 \pi}{12}}}{10} + \frac{3 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} e^{\frac{5 \pi}{12}}}{10}$$
-13*exp(pi/3)/80 - 3*sqrt(3)*exp(pi/3)/16 + 2*(-sqrt(2)/4 + sqrt(6)/4)^3*exp(5*pi/12)/5 + 3*(sqrt(2)/4 + sqrt(6)/4)^3*exp(5*pi/12)/10 + 3*(-sqrt(2)/4 + sqrt(6)/4)^2*(sqrt(2)/4 + sqrt(6)/4)*exp(5*pi/12)/5 + 3*(sqrt(2)/4 + sqrt(6)/4)^2*(-sqrt(2)/4 + sqrt(6)/4)*exp(5*pi/12)/10
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.