int(x^(3)dx)/((x^(2)+1)^(3//2))

Identify Integral: Identify the integral to be solved. We need to evaluate the integral ∫(x^3 dx) / ((x^2 + 1)^(3/2)).Use Substitution: Use substitution to simplify the integral. Let u = x^2 + 1. Then du = 2x dx, which implies (1/2)du = x dx.Rewrite in terms of u: Rewrite the integral in terms of u. Substitute x^2 = u - 1 into the integral and use the differential substitution from Step 2: ∫(x^3 dx) / ((x^2 + 1)^(3/2)) = ∫(x^2 * x dx) / ((x^2 + 1)^(3/2)) = ∫((u - 1) * (1/2)du) / (u^(3/2)) = (1/2) ∫((u - 1) / u^(3/2)) duSplit into simpler integrals: Split the integral into two simpler integrals. (1/2) ∫((u - 1) / u^(3/2)) du = (1/2) ∫(u / u^(3/2) - 1 / u^(3/2)) du = (1/2) ∫(u^(-1/2) - u^(-3/2)) duIntegrate each term: Integrate each term separately. (1/2) ∫(u^(-1/2) - u^(-3/2)) du = (1/2) [∫u^(-1/2) du - ∫u^(-3/2) du] = (1/2) [2u^(1/2) - 2u^(-1/2)/(-1)] = u^(1/2) + u^(-1/2)Substitute back in x: Substitute back in terms of x. Since u = x^2 + 1, we have: u^(1/2) + u^(-1/2) = (x^2 + 1)^(1/2) + 1/(x^2 + 1)^(1/2)Write final answer: Write the final answer. The integral ∫(x^3 dx) / ((x^2 + 1)^(3/2)) is equal to (x^2 + 1)^(1/2) + 1/(x^2 + 1)^(1/2) + C, where C is the constant of integration.

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Evaluate definite integrals using the chain rule

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\int \frac{x^{3} \, dx}{(x^{2}+1)^{\frac{3}{2}}}

Identify Integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral $\int \frac{x^3 \, dx}{(x^2 + 1)^{\frac{3}{2}}}$ .
Use Substitution: Use substitution to simplify the integral. $\newline$ Let $u = x^2 + 1$ . Then $du = 2x dx$ , which implies $(1/2)du = x dx$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . $\newline$ Substitute $x^2 = u - 1$ into the integral and use the differential substitution from Step $2$ : $\newline$ $\int\frac{x^3 dx}{(x^2 + 1)^{\frac{3}{2}}} = \int\frac{x^2 \cdot x dx}{(x^2 + 1)^{\frac{3}{2}}}$ $\newline$ $= \int\frac{(u - 1) \cdot \frac{1}{2}du}{u^{\frac{3}{2}}}$ $\newline$ $= \frac{1}{2} \int\frac{(u - 1)}{u^{\frac{3}{2}}} du$
Split into simpler integrals: Split the integral into two simpler integrals. $\newline$ (1/2) \int((u - 1) / u^{(3/2)}) du = (1/2) \int(u / u^{(3/2)} - 1 / u^{(3/2)}) du$\newline= (1/2) \int(u^{-1/2} - u^{-3/2}) du$
Integrate each term: Integrate each term separately. $\newline$ $(1/2) \int(u^{-1/2} - u^{-3/2}) du = (1/2) [\int u^{-1/2} du - \int u^{-3/2} du]$ $\newline$ $= (1/2) [2u^{1/2} - 2u^{-1/2}/(-1)]$ $\newline$ $= u^{1/2} + u^{-1/2}$
Substitute back in $x$ : Substitute back in terms of $x$ . $\newline$ Since $u = x^2 + 1$ , we have: $\newline$ $u^{(1/2)} + u^{(-1/2)} = (x^2 + 1)^{(1/2)} + \frac{1}{(x^2 + 1)^{(1/2)}}$
Write final answer: Write the final answer. $\newline$ The integral $\int \frac{x^3 \, dx}{(x^2 + 1)^{\frac{3}{2}}}$ is equal to $(x^2 + 1)^{\frac{1}{2}} + \frac{1}{(x^2 + 1)^{\frac{1}{2}}} + C$ , where $C$ is the constant of integration.