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

Problem

(∫_^)(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. Assign the variables for integration by parts by letting u=arctan(x) and d(v)=d(x)

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

4. Substitute these values into the integration by parts formula:

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

5. Evaluate the remaining integral using usubstitution, where w=1+x2 and d(w)=2*x*d(x) which implies x*d(x)=1/2*d(w)

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

6. Integrate the substitution result to get 1/2*ln(w) then substitute back w=1+x2

1/2*ln(1+x2)

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

Final Answer

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

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