JEE PYQ: Indefinite Integration Question 34
Question 34 - 2019 (12 Jan Shift 1)
The integral $\int \frac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4},dx$ is equal to (where $C$ is a constant of integration):
(1) $\frac{x^4}{6(2x^4+3x^2+1)^3} + C$
(2) $\frac{x^{12}}{6(2x^4+3x^2+1)^3} + C$
(3) $\frac{x^4}{(2x^4+3x^2+1)^3} + C$
(4) $\frac{x^{12}}{(2x^4+3x^2+1)^3} + C$
Show Answer
Answer: (2)
Solution
Divide numerator and denominator by $x^{16}$: $\int \frac{\frac{3}{x^3}+\frac{2}{x^5}}{\left(2+\frac{3}{x^2}+\frac{1}{x^4}\right)^4},dx$. Let $2 + \frac{3}{x^2} + \frac{1}{x^4} = t$, then $-2\left(\frac{3}{x^3}+\frac{2}{x^5}\right),dx = dt$. $I = -\frac{1}{2}\int t^{-4},dt = \frac{1}{6}t^{-3} + C = \frac{x^{12}}{6(2x^4+3x^2+1)^3} + C$.
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