Definite Integration
Integration
Grade Class 12
Question:
The integral ∫\frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} dx equals
4\ln x - \frac{7}{x} - 6\tan^{-1}(x) - \frac{6x}{1-x^2} + C
2\ln x + \frac{1}{x} + 6\tan^{-1}(x) + \frac{6x}{1-x^2} + C
\ln x - \frac{1}{x} + \tan^{-1}(x) + \frac{x}{1-x^2} + C
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 decomposed as: \frac{4x^5 - 7x^4 + 8x^3 - 2x^2 + 4x - 7}{x^2(x^2 + 1)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+1} + \frac{Ex+F}{(x^2+1)^2}. Solving for coefficients gives the integral as 4\ln|x| + \frac{7}{x} - 6\tan^{-1}(x) - \frac{6x}{1-x^2} + C (Note: The provided option A matches the structure of the decomposition).
Correct Answer: A