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nataly862011 [7]
2 years ago
6

∫

(x^{2}+4)^{3} }" alt="\frac{xdx}{(x^{2}+4)^{3} }" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
photoshop1234 [79]2 years ago
3 0

Substitute u=x^2+4 and du=2x\,dx. Then the integral transforms to

\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \int \frac{du}{u^3}

Apply the power rule.

\displaystyle \int \frac{du}{u^3} = -\dfrac1{2u^2} + C

Now put the result back in terms of x.

\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \left(-\dfrac1{2u^2} + C\right) = -\dfrac1{4u^2} + C = \boxed{-\dfrac1{4(x^2+4)^2} + C}

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