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\( \int \frac{x^{3}}{\left(x^{2}+1\right)^{3}} d x \) |
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Answer» \(\int \frac{x^3}{(x^2 + 1)^3 }dx = \int \frac{x^2.xdx}{(x^2 + 1)^3}\) \(1 + x^2 = t\) \(2xdx = dt\) \(xdx = \frac{dr}2\) \(\therefore \int \frac{x^3}{(x^2 + 1)^3}dx = \frac12 \int\frac{(t - 1)dt}{t^3}\) \( = \frac12 \int (t^{-2} - t^{-3} )dt\) \( =\frac12 \left(- \frac1t\ - \frac1{2t^2}\right)+ C\) \(= - \frac14 \left(\frac{2t + 1}{t^2}\right)+ C\) \(= - \frac14\left(\frac{2(1 + x^2)+1}{(1 +x^2)^2}\right)+ C\) \((\because t = 1 + x^2)\) \(= -\frac{2x^2 + 3}{4(1 + x^2)^2} + C\). |
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