$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$
A) $2 \arcsin 15$
B) $\frac { 2 } { \sqrt { 15 } } \arctan \sqrt { 15 }$
C) $\frac { 1 } { \sqrt { 15 } } \arcsin 15$
D) $15 \arctan 15$
E) $15 \arcsin \sqrt { 15 }$

Question

$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$ 
A) $2 \arcsin 15$
B) $\frac { 2 } { \sqrt { 15 } } \arctan \sqrt { 15 }$
C) $\frac { 1 } { \sqrt { 15 } } \arcsin 15$
D) $15 \arctan 15$
E) $15 \arcsin \sqrt { 15 }$

Quizplus · Accepted Answer

This integral can be solved by using the Weierstrass substitution, $t = \tan(\theta/2)$, which transforms the integral into a form that can be integrated using standard techniques. After performing the substitution and simplifying, the integral becomes a rational function of $t$, which can be integrated to yield the result $\frac { 2 } { \sqrt { 15 } } \arctan \sqrt { 15 }$. This method effectively handles the cosine term in the denominator and simplifies the integration process.

Evaluate the integral ∫0π/218−7cos⁡θdθ. \text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. } Evaluate the integral ∫0π/2​8−7cosθ1​dθ.

$\text { Evaluate the integral } \int _ { 0 } ^ { \pi / 2 } \frac { 1 } { 8 - 7 \cos \theta } d \theta \text {. }$