Use the substitution formula to evaluate the integral.
-$\int _ { 0 } ^ { \pi } ( 1 + \cos 5 t ) ^ { 2 } \sin 5 t d t$
A) $\frac { 1 } { 5 }$
B) $\frac { 8 } { 15 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 1 } { 15 }$

Question

Quizplus · Accepted Answer

Let $u = 1 + \cos 5t$, then $du = -5\sin 5t\ dt$.
Using the identity $\cos^2 x = \frac{1+\cos 2x}{2}$, we have
$$(1+\cos 5t)^2 = 1 + 2\cos 5t + \cos^2 5t = 2 + 2\cos 5t + \frac{1 + \cos 10t}{2} = 3 + 2\cos 5t + \frac{1}{2} u^2.$$
Therefore, the integral becomes
\begin{align*}
\int_0^\pi (1+\cos 5t)^2 \sin 5t\ dt &= \int_{u(0)}^{u(\pi)} \frac{-u^2}{2\cdot 25} \ du \
&= \left[ \frac{-u^3}{75} \right]_{u(0)}^{u(\pi)} \
&= \frac{8}{15}.
\end{align*}

Use the Substitution Formula to Evaluate the Integral ∫0π(1+cos⁡5t)2sin⁡5tdt\int _ { 0 } ^ { \pi } ( 1 + \cos 5 t ) ^ { 2 } \sin 5 t d t∫0π​(1+cos5t)2sin5tdt

Use the Substitution Formula to Evaluate the Integral $\int _ { 0 } ^ { \pi } ( 1 + \cos 5 t ) ^ { 2 } \sin 5 t d t$