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

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Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function (xtan^(-1)x)/((1+x^(2))^(3/2)) with respect to x. This is an indefinite integral, and we will look for a substitution that simplifies the integral.Choose Substitution: Choose a substitution. Let u = tan^(-1)x, which implies that x = tan(u). Then, we need to find dx in terms of du. To do this, we differentiate both sides with respect to u to get dx/du = sec^2(u). Therefore, dx = sec^2(u)du.Rewrite in terms of u: Rewrite the integral in terms of u. Substituting x = tan(u) and dx = sec^2(u)du into the integral, we get: int (xtan^(-1)x)/((1+x^(2))^(3/2))dx = int (tan(u) * u)/((1+tan^2(u))^(3/2)) * sec^2(u)du. Since 1 + tan^2(u) = sec^2(u), the integral simplifies to: int (tan(u) * u)/(sec^3(u)) * sec^2(u)du = int (tan(u) * u) * sec^(-1)(u)du.Simplify Further: Simplify the integral further. Since sec(u) = 1/cos(u) and tan(u) = sin(u)/cos(u), we can rewrite the integral as: int (sin(u)/cos(u) * u) * cos(u)du = int u * sin(u)du. Now we have an integral that can be solved using integration by parts.Apply Integration by Parts: Apply integration by parts. Let's use the formula for integration by parts: int u dv = uv - int v du. We choose u = u and dv = sin(u)du, which gives us du = du and v = -cos(u). Applying integration by parts, we get: int u * sin(u)du = -u * cos(u) - int (-cos(u))du = -u * cos(u) + int cos(u)du.Integrate cos(u): Integrate cos(u). The integral of cos(u) with respect to u is sin(u), so we have: int u * sin(u)du = -u * cos(u) + sin(u) + C, where C is the constant of integration.Convert to x: Convert back to x. We need to express our result in terms of x. Since u = tan^(-1)x and cos(u) = 1/sqrt(1+x^2), we have: -u * cos(u) + sin(u) + C = -tan^(-1)x * 1/sqrt(1+x^2) + sqrt(1+x^2) * 1/sqrt(1+x^2) + C. This simplifies to: -tan^(-1)x/sqrt(1+x^2) + 1 + C.Write Final Answer: Write the final answer. The integral of (xtan^(-1)x)/((1+x^(2))^(3/2)) with respect to x is: -tan^(-1)x/sqrt(1+x^2) + 1 + C.

$\int \frac{x\tan^{-1}(x)}{(1+x^{2})^{\frac{3}{2}}}dx$

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$\int \frac{x\tan^{-1}(x)}{(1+x^{2})^{\frac{3}{2}}}dx$