Evaluate the Integral integral of arctan(2x) with respect to x

Problem

(∫_^)(arctan(2*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(2*x) and d(v)=d(x)

3. Differentiate u to find d(u)=2/(1+(2*x)2)*d(x)=2/(1+4*x2)*d(x) and integrate d(v) to find v=x

4. Substitute these into the integration by parts formula to get x*arctan(2*x)−(∫_^)((2*x)/(1+4*x2)*d(x))

5. Evaluate the remaining integral using wsubstitution by letting w=1+4*x2 which implies d(w)=8*x*d(x) or 2*x*d(x)=1/4*d(w)

6. Substitute and integrate to get (∫_^)(1/(4*w)*d(w))=1/4*ln(w)

7. Combine the results and add the constant of integration C

Final Answer

(∫_^)(arctan(2*x)*d(x))=x*arctan(2*x)−1/4*ln(1+4*x2)+C

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