Find the indefinite integral. $$\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$$
A)$\frac { 2 \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 25 } + C$
B) $\frac { \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
C) $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
D) $\frac { \sqrt { \sin 5 \theta } \left( 3 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
E) $\frac { 2 \sqrt { \sin 5 \theta } \left( 3 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$

Question

Find the indefinite integral. $$\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$$
A)$\frac { 2 \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 25 } + C$ 
B) $\frac { \sqrt { \sin 5 \theta } \left( 5 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
C) $\frac { 2 \sqrt { \sin 5 \theta } \left( 5 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
D) $\frac { \sqrt { \sin 5 \theta } \left( 3 + \sin ^ { 2 } 5 \theta \right) } { 5 } + C$
E) $\frac { 2 \sqrt { \sin 5 \theta } \left( 3 - \sin ^ { 2 } 5 \theta \right) } { 5 } + C$

Quizplus · Accepted Answer

To solve the integral $\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$, we use the trigonometric identity $\cos^2(x) = 1 - \sin^2(x)$ to rewrite $\cos^3(5\theta)$ as $\cos(5\theta)(1 - \sin^2(5\theta))$. Then, we set $u = \sin(5\theta)$, so $du = 5\cos(5\theta)d\theta$, and rewrite the integral in terms of $u$, which simplifies the integration process. After integrating, we substitute back for $u$ to get the answer in terms of $\theta$, leading to the correct choice A.

Find the Indefinite Integral ∫cos⁡35θsin⁡5θdθ\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta∫sin5θ​cos35θ​dθ

Find the Indefinite Integral $\int \frac { \cos ^ { 3 } 5 \theta } { \sqrt { \sin 5 \theta } } d \theta$