$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$
A) $\frac { 1 } { 8 } \left( \ln \left| 4 + 5 \sin ( x ) + 4 \sin ^ { 2 } ( x ) \right| - \frac { 10 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) \right) + C$
B) $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \cos ( x ) + 5 } { \sqrt { 39 } } \right) + C$
C) $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \sin ( x ) + 5 - \sqrt { 39 } } { 8 \sin ( x ) + 5 + \sqrt { 39 } } \right| + C$
D) $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) + C$
E)$\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \cos ( x ) + 5 - \sqrt { 39 } } { 8 \cos ( x ) + 5 + \sqrt { 39 } } \right| + C$

Question

$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$ 
A) $\frac { 1 } { 8 } \left( \ln \left| 4 + 5 \sin ( x ) + 4 \sin ^ { 2 } ( x ) \right| - \frac { 10 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) \right) + C$
B) $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \cos ( x ) + 5 } { \sqrt { 39 } } \right) + C$
C) $\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \sin ( x ) + 5 - \sqrt { 39 } } { 8 \sin ( x ) + 5 + \sqrt { 39 } } \right| + C$
D) $\frac { 2 } { \sqrt { 39 } } \arctan \left( \frac { 8 \sin ( x ) + 5 } { \sqrt { 39 } } \right) + C$
E)$\frac { 1 } { \sqrt { 39 } } \ln \left| \frac { 8 \cos ( x ) + 5 - \sqrt { 39 } } { 8 \cos ( x ) + 5 + \sqrt { 39 } } \right| + C$

Quizplus · Accepted Answer

The integral can be solved by recognizing it as a standard form that involves a trigonometric function in the denominator. The correct answer involves the arctan function with a coefficient of $2/\sqrt{39}$ and the argument involving $\sin(x)$, which matches the form of option D. This is because the integral of a function that can be expressed in terms of $\sin(x)$ or $\cos(x)$ with a quadratic denominator often leads to an inverse trigonometric function, specifically $\arctan$, in the solution.

Use integration tables to find ∫cos⁡(x)4sin⁡2(x)+5sin⁡(x)+4dx\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x Use integration tables to find ∫4sin2(x)+5sin(x)+4cos(x)​dx

$\text { Use integration tables to find } \int \frac { \cos ( x ) } { 4 \sin ^ { 2 } ( x ) + 5 \sin ( x ) + 4 } d x$