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int(x^(2)dx)/((x^(2)+16)^(2))

$\int \frac{x^{2} d x}{\left(x^{2}+16\right)^{2}}$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int \frac{x^{2} d x}{\left(x^{2}+16\right)^{2}}

Substitution and Simplification: Simplify the integral using a substitution. $\newline$ Let $u = x^2 + 16$ , then $du = 2x dx$ . $\newline$ Rewrite $dx$ in terms of $du$ : $dx = \frac{du}{2x}$ . $\newline$ Substitute into the integral: $\newline$ $\int \frac{x^2 dx}{(x^2 + 16)^2} = \int \frac{x^2 \cdot (\frac{du}{2x})}{u^2}$ $\newline$ $= \frac{1}{2} \cdot \int \frac{du}{u^2}$
Further Simplification and Substitution: Simplify the integral further. $\newline$ The integral becomes: $\newline$ $(\frac{1}{2}) \cdot \int(\frac{du}{u^2}) = (\frac{1}{2}) \cdot (-\frac{1}{u}) + C$ $\newline$ $= -(\frac{1}{2u}) + C$ $\newline$ Substitute back for u: $\newline$ $= -(\frac{1}{2(x^2 + 16)}) + C$

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