Evaluate the integral by using multiple substitutions.
-$\int 3 \left( 3 x ^ { 2 } - 5 \right) \sin ^ { 5 } \left( x ^ { 3 } - 5 x \right) \cos \left( x ^ { 3 } - 5 x \right) d x$
A) $15 \sin ^ { 4 } \left( x ^ { 3 } - 5 x \right) + C$
B) $\frac { 1 } { 2 } \cos ^ { 6 } \left( 3 x ^ { 2 } \right) + C$
C) $\frac { 1 } { 2 } \sin ^ { 6 } \left( x ^ { 3 } - 5 x \right) + C$
D) $2 \sin ^ { 6 } \left( x ^ { 3 } - 5 x \right) + C$

Question

Quizplus · Accepted Answer

We can use the following substitutions:
$u = x^3 - 5x$
$du = (3x^2 - 5) dx$
$v = \sin^2 u$
$dv = 2\sin u \cos u du$
Using the chain rule, we can rewrite the integral as follows:
$\int 3\left(3x^2 - 5\right) \sin^5(x^3 - 5x) \cos (x^3 - 5x) dx = \int 3v^2 dv$
This integral is straightforward to solve:
$\int 3v^2 dv = \frac{1}{2}v^3 + C = \frac{1}{2}\sin^6(x^3 - 5x) + C$
Therefore, the correct answer is C.

Evaluate the Integral by Using Multiple Substitutions ∫3(3x2−5)sin⁡5(x3−5x)cos⁡(x3−5x)dx\int 3 \left( 3 x ^ { 2 } - 5 \right) \sin ^ { 5 } \left( x ^ { 3 } - 5 x \right) \cos \left( x ^ { 3 } - 5 x \right) d x∫3(3x2−5)sin5(x3−5x)cos(x3−5x)dx

Evaluate the Integral by Using Multiple Substitutions $\int 3 \left( 3 x ^ { 2 } - 5 \right) \sin ^ { 5 } \left( x ^ { 3 } - 5 x \right) \cos \left( x ^ { 3 } - 5 x \right) d x$