Evalúa las integrales definidas

No aparece la variable de integración, la nombraremos x.

Hagamos un truquillo

$$\begin{align}&\int_{\pi/4}^{\pi/3}sen^5xdx =\int_{\pi/4}^{\pi/3}sen^2x·sen^2x·senx dx=\\ &\\ &\\ &\int_{\pi/4}^{\pi/3}(1-\cos^2x)(1-\cos^2x)senxdx=\\ &\\ &\\ &\int_{\pi/4}^{\pi/3}(1-2cos^2x+\cos^4x)senxdx =\\ &\\ &t=cosx\quad \quad dt=-senxdx\\ &x=\pi/4 \implies t = \sqrt 2/2\\ &x=\pi/3 \implies t = 1/2\\ &\\ &=-\int_{\sqrt 2/2}^{1/2}(1-2t^2+t4)dt =\\ &\\ &\\ &- \left[t -\frac{2t^3}{3}+\frac{t^5}{5}  \right]_{\sqrt 2/2}^{1/2}=\\ &\\ &-\left(\frac 12- \frac{1}{12}+\frac{1}{160}-\frac{\sqrt{2}}{2}+\frac{\sqrt 2}{6}-\frac{\sqrt 2}{40}  \right)=\\ &\\ &-\left( \frac{240-40+3-240 \sqrt 2+80 \sqrt 2-12 \sqrt 2}{480} \right)=\\ &\\ &\\ &-\left(\frac{203-172 \sqrt 2}{480}\right)=\\ &\\ &\\ &\frac{172 \sqrt 2-203}{480}\approx 0.08384319318\\ &\\ &\\ &\end{align}$$

Y eso es todo.