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Find:\( \int \frac{2 x}{\left(x^{2}+1\right) \times\left(x^{2}+2\right)^{2}} d x \) |
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Answer» \( I = \int\frac{2x}{(x^2 + 1)(x^2 + 2)^2}dx\) Let x2 = t 2x dx = dt \(\therefore I = \int \frac{dt}{(t+1)(t+2)^2}\) Let \(\frac{1}{(t + 1)(t + 2)^2} = \frac{A}{t+1}+ \frac{B}{t+2}+\frac{C}{(t+2)^2}\) ⇒ I = A (t + 2)2 + B(t +1) (t + 2) + c(t +1) Put t = -1, we get A = 1 Put t = -2, we get C = -1 Put t = 0, we get 4A + 2B + C = 1 ⇒ 2B + 4 - 1 = 1 ⇒ 2B = -2 ⇒ B = -1 \(\therefore \frac{1}{(t +1)(t +2)^2}= \frac1{t+1}-\frac1{t+2}-\frac1{(t+2)^2}\) \(\therefore I = \int \left(\frac{1}{t +1}- \frac1{t+ 2}-\frac1{(t + 2)^2}\right)dt = log(t +1) - log (t +2) + \frac1{t+2}+C\) \(= log \left|\frac{t+1}{t+2}\right|+ \frac1 {t+2}+C \) \(= \log\left|\frac{x^2 +1}{x^2 +2 }\right|+\frac{1}{x^2+2}+C\) |
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