$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$

A) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
B) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } + \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
C) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$
D) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
E) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$

Question

$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$

A) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
B) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } + \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
C) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$
D) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } + ( x + 5 ) \right| + C$
E) $\int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x = \sqrt { x ^ { 2 } + 10 x + 30 } - 5 \ln \left| \sqrt { x ^ { 2 } + 10 x + 30 } \right| + C$

Quizplus · Accepted Answer

Completing the square gives $x^2+10x+30=(x+5)^2+5$. Substituting $u=x+5$ gives $\int\frac{u-5}{\sqrt{u^2+5}}\,du$. The expression inside the square root is of the form $u^2+a^2$, so we substitute $u=a\tan\theta$ to get $\int\,d\theta=\theta+C$. Undoing the substitution, $\theta=\arctan\frac{x+5}{\sqrt{5}}$, so 
$$\int\frac{x}{\sqrt{x^2+10x+30}}\,dx=\arctan\frac{x+5}{\sqrt{5}}-\ln\left|\sqrt{x^2+10x+30}+(x+5)\right|+C.$$
The answer is $\boxed{\textbf{(A)}}$.

Complete the square and find ∫xx2+10x+30dx. \text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. } Complete the square and find ∫x2+10x+30​x​dx.

$\text { Complete the square and find } \int \frac { x } { \sqrt { x ^ { 2 } + 10 x + 30 } } d x \text {. }$