Integrate. intsqrtx*arctan sqrtxdx

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Recognize the integral: Recognize the integral we need to solve. We have the integral of the square root of x times the arc tangent of the square root of x, which is written as: ∫ sqrt(x) * arctan(sqrt(x)) dxUse integration by parts: Use integration by parts. Integration by parts is given by the formula: ∫ u dv = uv - ∫ v du We need to choose u and dv such that du and v are easier to integrate. Let's choose: u = arctan(sqrt(x)) (since its derivative is simpler) dv = sqrt(x) dx (since its antiderivative is straightforward)Compute du and v: Compute du and v. To find du, we differentiate u with respect to x: du = d/dx [arctan(sqrt(x))] dx Using the chain rule and the derivative of arctan(x), which is 1/(1+x^2), we get: du = (1/2) * (1/(1+x)) * (1/sqrt(x)) dx Now, to find v, we integrate dv: v = ∫ sqrt(x) dx v = (2/3) * x^(3/2)Apply integration by parts formula: Apply the integration by parts formula. Now we have all the parts needed to apply the integration by parts formula: ∫ sqrt(x) * arctan(sqrt(x)) dx = uv - ∫ v du = arctan(sqrt(x)) * (2/3) * x^(3/2) - ∫ (2/3) * x^(3/2) * (1/2) * (1/(1+x)) * (1/sqrt(x)) dxSimplify the integral: Simplify the integral. Simplify the expression inside the integral: ∫ (2/3) * x^(3/2) * (1/2) * (1/(1+x)) * (1/sqrt(x)) dx = ∫ (1/3) * x * (1/(1+x)) dx Now we have a simpler integral to solve.Solve the simplified integral: Solve the simplified integral. To solve ∫ (1/3) * x * (1/(1+x)) dx, we can use a simple substitution: Let t = 1 + x, then dt = dx and x = t - 1. Our integral becomes: ∫ (1/3) * (t - 1) * (1/t) dt = (1/3) * ∫ (1 - 1/t) dt = (1/3) * (t - ln|t|) Now we substitute back for x: = (1/3) * (1 + x - ln|1 + x|)Write the final answer: Write the final answer. Combining the result from integration by parts with the solved integral, we get: ∫ sqrt(x) * arctan(sqrt(x)) dx = arctan(sqrt(x)) * (2/3) * x^(3/2) - (1/3) * (1 + x - ln|1 + x|) + C where C is the constant of integration.

Integrate. $\newline$ $\int \sqrt{x} \cdot \operatorname{arctan} \sqrt{x} d x$

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Integrate. $\newline$ $\int \sqrt{x} \cdot \operatorname{arctan} \sqrt{x} d x$