Evaluate the integral. $$\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$$
A) $\frac { 1 } { 2 } \left( \tan ( x + 1 ) + \frac { 17 } { x ^ { 2 } + 2 x + 2 } \right) + C$
B) $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 2 ) + \frac { 7 } { x ^ { 2 } + 2 } \right) + C$
C) $\frac { 1 } { 2 } \left( \tan ^ { - 1 } ( x + 7 ) + \frac { 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$
D) $\frac { 7 } { 2 } \left( \tan ( x + 2 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$
E) $\frac { 7 } { 2 } \left( \tan ^ { - 1 } ( x + 1 ) + \frac { x + 1 } { x ^ { 2 } + 2 x + 2 } \right) + C$

Question

Quizplus · Accepted Answer

The integral involves a quadratic expression in the denominator raised to the power of 2, which suggests a partial fraction decomposition or a substitution that simplifies the expression. The correct answer involves recognizing that the expression inside the integral can be related to the derivative of the arctangent function, $\tan^{-1}(x)$, after completing the square in the denominator to match the form $\frac{1}{1+x^2}$, and adjusting the numerator accordingly to facilitate this substitution. Option E correctly applies this principle, using $\tan^{-1}(x+1)$ to account for the completed square and the linear term in the numerator to match the derivative of the arctangent function, making it the correct choice.

Evaluate the Integral ∫7dx(x2+2x+2)2\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }∫(x2+2x+2)27dx​

Evaluate the Integral $\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$