Evaluate the integral.
$\int 5 ^ { \sin \theta } \cos \theta d \theta$
A) $\frac { 5 ^ { \cos \theta } } { \ln ( 5 ) } + C$
B) $\frac { 5 ^ { \sin \theta } } { \ln ( \sin \theta ) } + C$
C) $\frac { 5 ^ { \sin \theta } } { \ln ( 5 ) } + C$
D) $\frac { 5 ^ { \cos \theta } } { \ln ( \sin \theta ) }$
E) none of these

Question

Quizplus · Accepted Answer

Using the substitution $u = \sin\theta$, we get $du = \cos\theta d\theta$, and the integral becomes
$$\int 5^{\sin\theta}\cos\theta d\theta = \int 5^u du.$$
Now, we can simply integrate this to get
$$\int 5^u du = \frac{5^u}{\ln 5}+C = \frac{5^{\sin\theta}}{\ln 5}+C.$$
Therefore, the correct choice is (C).

Evaluate the Integral ∫5sin⁡θcos⁡θdθ\int 5 ^ { \sin \theta } \cos \theta d \theta∫5sinθcosθdθ

Evaluate the Integral $\int 5 ^ { \sin \theta } \cos \theta d \theta$