Definite Integration
Integration of rational functions
Grade Class 12
Question:
∫ <sup>x<sup>2</sup> + 3</sup> / <sub>x<sup>6</sup>(x<sup>2</sup> + 1)</sub> dx equals
C - <sup>2</sup>/<sub>x</sub> + <sup>2</sup>/<sub>3x<sup>3</sup></sub> - <sup>3</sup>/<sub>5x<sup>5</sup></sub> - 2tan<sup>-1</sup> x
C - <sup>2</sup>/<sub>x<sup>2</sup></sub> + <sup>2</sup>/<sub>3x<sup>4</sup></sub> - <sup>3</sup>/<sub>5x<sup>6</sup></sub> - 2tan<sup>-1</sup> x
C - <sup>2</sup>/<sub>x</sub> + <sup>2</sup>/<sub>3x<sup>3</sup></sub> - <sup>3</sup>/<sub>5x<sup>5</sup></sub> + 2tan<sup>-1</sup> x
C - <sup>2</sup>/<sub>x<sup>2</sup></sub> + <sup>2</sup>/<sub>3x<sup>4</sup></sub> + <sup>3</sup>/<sub>5x<sup>6</sup></sub> + 2tan<sup>-1</sup> x
Step-by-Step Solution
Key Concept: The integral can be solved by expressing the integrand as (x^2 + 3) / (x^6(x^2 + 1)) = (A/x^6 + B/x^4 + C/x^2) + D/(x^2 + 1) or by manipulating the numerator to match the denominator's structure, specifically using the substitution x^2 = t or dividing numerator and denominator by x^6.
The integrand is (x^2 + 3) / (x^6(x^2 + 1)). We can write x^2 + 3 = (x^2 + 1) + 2. So the integral becomes \int (x^2 + 1) / (x^6(x^2 + 1)) dx + \int 2 / (x^6(x^2 + 1)) dx = \int x^-6 dx + 2 \int 1 / (x^6(x^2 + 1)) dx. The first part is -1/(5x^5). For the second part, use partial fractions or the identity 1/(x^2(x^2+1)) = 1/x^2 - 1/(x^2+1). Repeatedly applying this or using the substitution x = 1/t leads to the result.
Correct Answer: 1