Definite Integration
Integration of rational functions
Grade Class 12
Question:
18. $\int \frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} dx$ equals
(A) $4\ln x - \frac{7}{x} - 6\tan^{-1}(x) - \frac{6x}{1-x^2} + C$
(B) $2\ln x + \frac{1}{x} + 6\tan^{-1}(x) + \frac{6x}{1-x^2} + C$
(C) $\ln x - \frac{1}{x} + \tan^{-1}(x) + \frac{x}{1-x^2} + C$
(D) $4\ln x + \frac{7}{x} + 6\tan^{-1}(x) + \frac{6x}{1+x^2} + C$
Step-by-Step Solution
Key Concept: Perform partial fraction decomposition on the integrand. The integrand can be split into terms involving 1/x, 1/x^2, and terms of the form (Ax+B)/(x^2+1) and (Cx+D)/(x^2+1)^2.
The integrand is $\frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2}$. By partial fraction decomposition, this can be written as $\frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+1} + \frac{Ex+F}{(x^2+1)^2}$. Solving for the coefficients, we integrate term by term to obtain the result.
Correct Answer: 4