Evaluate the integral.
-$$\int 7 \cos ^ { 3 } 5 x d x$$
A) $\frac { 7 } { 5 } \sin 5 x - \frac { 7 } { 15 } \cos ^ { 3 } 5 x + C$
B) $\frac { 7 } { 5 } \sin 5 x + \frac { 7 } { 15 } \sin ^ { 3 } 5 x + C$
C) $7 \sin 5 x - \frac { 7 } { 3 } \sin ^ { 3 } 5 x + C$
D) $\frac { 7 } { 5 } \sin 5 x - \frac { 7 } { 15 } \sin ^ { 3 } 5 x + C$

Question

Quizplus · Accepted Answer

We can use the trigonometric identity $\cos^3x=\cos x(1-\sin^2x)$ and substitute $u=5x$ to get:
\begin{align*}
\int7\cos^35x\,dx&=\int7\cos 5x(1-\sin^25x)\,dx\
&=\int7\cos 5x\,dx-\int7\sin^25x\cos 5x\,dx\
&=\frac{7}{5}\sin 5x-\frac{7}{15}\sin^35x+C\qquad	ext{(using }u=\sin 5x	ext{)}\
&=\frac{7}{5}\sin 5x-\frac{7}{15}\sin^35x+C.
\end{align*} 
Thus, the correct answer is D.

Evaluate the Integral ∫7cos⁡35xdx\int 7 \cos ^ { 3 } 5 x d x∫7cos35xdx

Evaluate the Integral $\int 7 \cos ^ { 3 } 5 x d x$