Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 2 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } + 3 \right) ( x - 5 ) } d x$$
A) $2 \ln | x - 5 | + \frac { \sqrt { 3 } } { 3 } \tan ^ { - 1 } \left( \frac { x \sqrt { 3 } } { 3 } \right) + C$
B) $2 \ln | x - 5 | + \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { x } { 3 } \right) + C$
C) $2 \ln | x - 5 | + \tan ^ { - 1 } \left( \frac { x \sqrt { 3 } } { 3 } \right) + C$
D) $\ln | x - 5 | + \frac { \sqrt { 3 } } { 3 } \tan ^ { - 1 } \left( \frac { x \sqrt { 3 } } { 3 } \right) + C$

Question

Quizplus · Accepted Answer

To solve the integral, we first express the integrand as a sum of partial fractions. Given the denominator $(x^2 + 3)(x - 5)$, we set up the partial fraction decomposition as follows:$$\frac{2x^2 + x + 1}{(x^2 + 3)(x - 5)} = \frac{Ax + B}{x^2 + 3} + \frac{C}{x - 5}$$Multiplying both sides by the common denominator $(x^2 + 3)(x - 5)$ and equating coefficients or setting specific values for $x$ to solve for $A$, $B$, and $C$, we find the values of $A$, $B$, and $C$ that satisfy the equation. After finding $A$, $B$, and $C$, we integrate each term separately.The integral of $\frac{1}{x^2 + a^2}$ is $\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$, and the integral of $\frac{1}{x - a}$ is $\ln|x - a| + C$. Applying these rules, we get the integral in the form of choice A, which includes a logarithmic term for the linear denominator and an arctan term for the quadratic denominator, correctly adjusted for the coefficients found from the partial fraction decomposition.

Express the Integrand as a Sum of Partial Fractions and Evaluate