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\( \int \frac{x^{7} d x}{\left(1+x^{2}\right)^{5}} \)integration x⁷/(1+x²)⁵ |
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Answer» Let I = \(\int\frac{x^7dx}{(1+x^2)^5}\) Put 1 + x2 = t ⇒ 2xdx = dt \(\therefore\) I = \(\frac12\int\frac{(t-1)^3dt}{t^5}\) = \(\frac12\int\frac{t^3-3t^2+3t-1}{t^5}dt\) = \(\frac12\int[t^{-2}-3t^{-3} +3t^{-4}-t^{-5}]dt\) = \(\frac12\left[\frac{-1}{t}+\frac32\frac1{t^2}-\frac33\frac1{t^3}+\frac14\frac1{t^4}\right]+C\) = \(\frac12\left[\frac{-1}{1+x^2}+\frac3{2(1+x^2)^2}-\frac{1}{(1+x^2)^3}+\frac14\frac{1}{(1+x^2)^4}\right]+C\) |
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