Please show the integration of $\frac{x^{3}}{{(x^{4} + 7)}^{8}} d x$ $(x^3)/((x^4+7)^8)dx$ step by step?

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Please show the integration of $\frac{x^{3}}{{(x^{4} + 7)}^{8}} d x$ $(x^3)/((x^4+7)^8)dx$ step by step?

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Answer 1 · 2018-05-23T04:55:56

We have

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

Let $u = x^{4} + 7 d x$ $u = x^4 + 7dx$ . Then $d u = 4 x^{3} d x$ $du = 4x^3 dx$ and $d x = \frac{d u}{4 x^{3}}$ $dx = (du)/(4x^3)$

$I = \int \frac{x^{3}}{u^{8}} \cdot \frac{d u}{4 x^{3}}$ $I = int x^3/u^8 * (du)/(4x^3)$

$I = \frac{1}{4} \int u^{- 8} d u$ $I = 1/4int u^-8 du$

$I = \frac{1}{4} (- \frac{1}{7} u^{- 7}) + C$ $I = 1/4(-1/7u^-7) + C$

$I = - \frac{1}{28} {(x^{4} + 7)}^{- 7} + C$ $I = -1/28(x^4 + 7)^-7 +C$

Hopefully this helps!

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Please show the integration of $\frac{x^{3}}{{(x^{4} + 7)}^{8}} d x$ $(x^3)/((x^4+7)^8)dx$ step by step?

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