Evaluate the integral by using multiple substitutions.
-$\int \sqrt { 8 + \sin ^ { 2 } ( x - 9 ) } \sin ( x - 9 ) \cos ( x - 9 ) d x$
A) $\frac { 3 } { 4 } \sqrt { 8 + \sin ^ { 2 } ( x - 9 ) } + C$
B) $\frac { 1 } { 3 } \left( 8 + \sin ^ { 2 } ( x - 9 ) \right) ^ { 3 / 2 } + C$
C) $\left( 8 + \cos ^ { 2 } ( x - 9 ) \right) ^ { 3 / 2 } + C$
D) $\frac { 1 } { 3 } \left( 8 + \sin ^ { 2 } x \right) ^ { 3 / 2 } + C$

Question

Quizplus · Accepted Answer

First, let $u = x - 9$, which simplifies the integral to $\int \sqrt{8 + \sin^2(u)} \sin(u) \cos(u) du$. Then, let $v = \sin^2(u)$, so $dv = 2\sin(u)\cos(u)du$, which simplifies the integral further to $\frac{1}{2}\int \sqrt{8 + v} dv$. Integrating this gives $\frac{1}{3}(8 + v)^{3/2} + C$, and substituting back for $v$ and then $u$ gives $\frac{1}{3}(8 + \sin^2(x - 9))^{3/2} + C$.

Evaluate the Integral by Using Multiple Substitutions ∫8+sin⁡2(x−9)sin⁡(x−9)cos⁡(x−9)dx\int \sqrt { 8 + \sin ^ { 2 } ( x - 9 ) } \sin ( x - 9 ) \cos ( x - 9 ) d x∫8+sin2(x−9)​sin(x−9)cos(x−9)dx

Evaluate the Integral by Using Multiple Substitutions $\int \sqrt { 8 + \sin ^ { 2 } ( x - 9 ) } \sin ( x - 9 ) \cos ( x - 9 ) d x$