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int2x(x^(2)+1)^(2)dx=

$\int 2 x\left(x^{2}+1\right)^{2} d x=$

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Evaluate definite integrals using the power rule

Full solution

Q.

\int 2 x\left(x^{2}+1\right)^{2} d x=

Given Integral: We are given the integral to evaluate: $\newline$ $\int 2x(x^2 + 1)^2 \, dx$ $\newline$ To solve this integral, we will use substitution. Let's choose $u = x^2 + 1$ , then $du = 2x \, dx$ .
Substitution: Now we substitute $u$ into the integral and also express $dx$ in terms of $du$ : $\int 2x(x^2 + 1)^2 dx = \int u^2 du$
Integral of $u^2$ : Next, we find the integral of $u^2$ with respect to $u$ : $\newline$ $\int u^2 \, du = \frac{1}{3}u^3 + C$ , where $C$ is the constant of integration.
Substitute back to x: We substitute back the original variable x to get the answer in terms of x: $\frac{1}{3}u^3 + C = \frac{1}{3}(x^2 + 1)^3 + C$
Final Answer: Now we have the final answer: $\newline$ $\int 2x(x^2 + 1)^2 \, dx = \frac{1}{3}(x^2 + 1)^3 + C$

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