Definite Integration
Indefinite Integration
Grade Class 12
Question:
The evaluation of $\int \frac{p x^{p+2q-1} - q x^{q-1}}{x^{2p+2q} + 2x^{p+q} + 1} dx$ is
(A) $-\frac{x^p}{x^{p+q} + 1} + C$
(B) $\frac{x^q}{x^{p+q} + 1} + C$
(C) $-\frac{x^q}{x^{p+q} + 1} + C$
(D) $\frac{x^p}{x^{p+q} + 1} + C$
Step-by-Step Solution
Key Concept: Divide numerator and denominator by x^(2p+q) or manipulate the expression to identify the derivative of a function in the denominator.
Let $I = \int \frac{p x^{p+2q-1} - q x^{q-1}}{(x^{p+q} + 1)^2} dx$. Divide numerator and denominator by $x^{2p+2q}$ or substitute $u = x^{p+q} + 1$. Alternatively, notice that the derivative of $\frac{x^q}{x^{p+q}+1}$ relates to the integrand.
Correct Answer: C