Evaluate the Integral integral of xarctan(x) with respect to x

Problem

(∫_^)(x*arctan(x)*d(x))

Solution

1. Identify the method of integration by parts, using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the parts for substitution by letting u=arctan(x) and d(v)=x*d(x)

3. Differentiate u to find d(u)=1/(1+x2)*d(x)

4. Integrate d(v) to find v=(x2)/2

5. Substitute these into the integration by parts formula:

(∫_^)(x*arctan(x)*d(x))=(x2)/2*arctan(x)−(∫_^)((x2)/(2*(1+x2))*d(x))

6. Simplify the integral on the right by adding and subtracting 1 in the numerator:

1/2*(∫_^)((x2)/(1+x2)*d(x))=1/2*(∫_^)((x2+1−1)/(1+x2)*d(x))

7. Split the fraction into two parts:

1/2*(∫_^)((1−1/(1+x2))*d(x))

8. Integrate the terms inside the parentheses:

1/2*(x−arctan(x))

9. Combine all parts and add the constant of integration C

(x2)/2*arctan(x)−1/2*x+1/2*arctan(x)+C

10. Factor the expression to simplify the final form:

(x2+1)/2*arctan(x)−x/2+C

Final Answer

(∫_^)(x*arctan(x)*d(x))=(x2+1)/2*arctan(x)−x/2+C

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