\( \int \frac{x^{3}}{\left(x^{4}+7\right)^{8}} d x= \)

1.	\( \int \frac{x^{3}}{\left(x^{4}+7\right)^{8}} d x= \)
Answer» Let \(I = \int \frac{x^3}{(x^4 + 7)^8}dx\) Let x⁴ + 7 = t 4x³dx = dt \(\therefore I = \frac14\int\frac{dt}{t^8} = \frac14 \times \frac{-1}{7}\left[\frac1{t^7}\right] + C\) \(= \frac{-1}{28}\frac{1}{(x^4 + 7)^7} +C\) \((\because t = x^4 + 7)\)

Answer»

Let \(I = \int \frac{x^3}{(x^4 + 7)^8}dx\)

Let x⁴ + 7 = t

4x³dx = dt

\(\therefore I = \frac14\int\frac{dt}{t^8} = \frac14 \times \frac{-1}{7}\left[\frac1{t^7}\right] + C\)

\(= \frac{-1}{28}\frac{1}{(x^4 + 7)^7} +C\) \((\because t = x^4 + 7)\)

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\( \int \frac{x^{3}}{\left(x^{4}+7\right)^{8}} d x= \)

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