JEE PYQ: Indefinite Integration Question 3
Question 3 - 2021 (18 Mar Shift 1)
If $f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2},dx$, $(x \ge 0)$, $f(0) = 0$ and $f(1) = \frac{1}{K}$, then the value of K is ______.
Show Answer
Answer: (4)
Solution
Rewrite as $\int \frac{5x^{-6} + 7x^{-8}}{(x^{-5} + x^{-7} + 2)^2},dx$. Let $x^{-5} + x^{-7} + 2 = t$, then $(-5x^{-6} - 7x^{-8}),dx = dt$. So $f(x) = \int \frac{-dt}{t^2} = \frac{1}{t} + c = \frac{x^7}{x^2 + 1 + 2x^7} + c$. $f(0) = 0 \Rightarrow c = 0$. $f(1) = \frac{1}{4}$, so $K = 4$.
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