Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 16 } } d x$ .
A) $\frac { 2 } { 3 } ( x + 16 ) ^ { 3 / 2 } ( x - 32 ) + C$
B) $\frac { 2 } { 3 } ( x - 16 ) ^ { 1 / 2 } ( x + 32 ) + C$
C) $\frac { 2 } { 3 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$
D) $\frac { 3 } { 2 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$

Question

Quizplus · Accepted Answer

To solve $\int \frac { x } { \sqrt { x + 16 } } d x$ using integration by parts, we let $u = x$ which implies $du = dx$, and $dv = \frac{1}{\sqrt{x+16}}dx$ which implies $v = 2\sqrt{x+16}$. Applying the integration by parts formula $\int u dv = uv - \int v du$, we get $x(2\sqrt{x+16}) - \int 2\sqrt{x+16} dx$. Further simplification and integration lead to the expression $\frac { 2 } { 3 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$, which matches choice C.

Use Integration by Parts to Evaluate the Integral ∫xx+16dx\int \frac { x } { \sqrt { x + 16 } } d x∫x+16​x​dx

Use Integration by Parts to Evaluate the Integral $\int \frac { x } { \sqrt { x + 16 } } d x$