Find the indefinite integral. $\int \cos ^ { 4 } 5 x d x$
A) $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$
B) $\frac { 60 x + 8 \sin ( 5 ) x + \sin ( 20 ) x } { 160 } + C$
C) $\frac { 20 x + 4 \sin ( 10 ) x + \sin ( 15 ) x } { 160 } + C$
D) $\frac { 60 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$
E) $\frac { 20 x + 8 \sin ( 10 ) x + \sin ( 20 ) x } { 160 } + C$

Question

Quizplus · Accepted Answer

We can use the reduction formula for $\cos^n(x)$ to simplify the integral:
\begin{align*}
\int \cos^4(5x)dx &= \int \cos^2(5x)\cos^2(5x)dx \
&= \int \left(\frac{1+\cos(10x)}{2}ight)\left(\frac{1+\cos(10x)}{2}ight)dx \
&= \frac{1}{4}\int (1+\cos(10x)+\cos^2(10x))dx \
&= \frac{1}{4}\left(x+\frac{1}{10}\sin(10x)+\frac{1}{20}x+\frac{1}{40}\sin(20x)ight)+C \
&= \frac{3}{20}x+\frac{1}{40}\sin(20x)+\frac{1}{8}\sin(10x)+C
\end{align*}
Thus, the correct choice is $\boxed{	extbf{(D)}}$.

Find the Indefinite Integral ∫cos⁡45xdx\int \cos ^ { 4 } 5 x d x∫cos45xdx

Find the Indefinite Integral $\int \cos ^ { 4 } 5 x d x$