Definite Integration
Integration
Grade Class 12
Question:
The evaluation of ∫ <sup>p x<sup>p+2q-1</sup> - q x<sup>q-1</sup></sup> ⁄ <sub>x<sup>2p+2q</sup> + 2x<sup>p+q</sup> + 1</sub> dx is
(A) - <sup>x<sup>p</sup></sup> ⁄ <sub>x<sup>p+q</sup> + 1</sub> + C
(B) <sup>x<sup>q</sup></sup> ⁄ <sub>x<sup>p+q</sup> + 1</sub> + C
(C) - <sup>x<sup>q</sup></sup> ⁄ <sub>x<sup>p+q</sup> + 1</sub> + C
(D) <sup>x<sup>p</sup></sup> ⁄ <sub>x<sup>p+q</sup> + 1</sub> + 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 of x^(p+q). Specifically, notice that the denominator is (x^(p+q) + 1)^2. Rewrite the numerator to match the derivative of the denominator's base.
Let I = ∫ (p x<sup>p+2q-1</sup> - q x<sup>q-1</sup>) / (x<sup>p+q</sup> + 1)<sup>2</sup> dx. Divide numerator and denominator by x<sup>p+q-1</sup>. Alternatively, let u = x<sup>p+q</sup> + 1, then du = (p+q)x<sup>p+q-1</sup> dx. This is a standard substitution problem for rational functions.
Correct Answer: 3