Find the Antiderivative arctan(x)

Problem

(∫_^)(arctan(x)*d(x))

Solution

1. Identify the method of integration by parts, which uses 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 into the integration by parts formula.

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

5. Apply a substitution to solve the remaining integral by letting w=1+x2 which implies d(w)=2*x*d(x) or x*d(x)=1/2*d(w)

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

6. Integrate the substituted expression to get 1/2*ln(w)

1/2*ln(w)=1/2*ln(1+x2)

7. Combine the results 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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