Evaluate the integral by using a substitution prior to integration by parts.
-$$\int x ^ { 2 } \sqrt { x + 5 } d x$$
A) $\frac { \left( 15 x ^ { 2 } - 60 x + 200 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$
B) $\frac { \left( 30 x ^ { 2 } - 120 x + 80 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$
C) $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) \sqrt { ( x + 5 ) } } { 105 } + C$
D) $\frac { \left( 30 x ^ { 2 } - 120 x + 400 \right) ( x + 5 ) ^ { 3 / 2 } } { 105 } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int x^2 \sqrt{x + 5} dx$, we use the substitution $u = x + 5$, which implies $du = dx$ and $x = u - 5$. Substituting these into the integral gives us an integral in terms of $u$, which we can then solve using integration by parts, if necessary. After solving, we substitute back $u = x + 5$ to get the answer in terms of $x$. The correct solution involves these steps and matches option D.

Evaluate the Integral by Using a Substitution Prior to Integration ∫x2x+5dx\int x ^ { 2 } \sqrt { x + 5 } d x∫x2x+5​dx

Evaluate the Integral by Using a Substitution Prior to Integration $\int x ^ { 2 } \sqrt { x + 5 } d x$