Evaluate the Integral integral of arctan(4x) with respect to x

Problem

(∫_^)(arctan(4*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(4*x) and d(v)=d(x)

3. Differentiate u to find d(u)=4/(1+(4*x)2)*d(x)=4/(1+16*x2)*d(x)

4. Integrate d(v) to find v=x

5. Substitute these into the integration by parts formula to get x*arctan(4*x)−(∫_^)((4*x)/(1+16*x2)*d(x))

6. Apply a substitution for the remaining integral by letting w=1+16*x2 which implies d(w)=32*x*d(x) or 4*x*d(x)=1/8*d(w)

7. Evaluate the new integral (∫_^)(1/(8*w)*d(w))=1/8*ln(w)

8. Substitute back w=1+16*x2 and add the constant of integration C

Final Answer

(∫_^)(arctan(4*x)*d(x))=x*arctan(4*x)−1/8*ln(1+16*x2)+C

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