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

Problem

(∫_^)(arctan(4*t)*d(t))

Solution

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

2. Assign the variables for integration by parts by letting u=arctan(4*t) and d(v)=d(t)

3. Differentiate u to find d(u)=4/(1+(4*t)2)*d(t)=4/(1+16*t2)*d(t)

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

5. Substitute these into the integration by parts formula to get t*arctan(4*t)−(∫_^)((4*t)/(1+16*t2)*d(t))

6. Apply a usubstitution for the remaining integral by letting w=1+16*t2 which implies d(w)=32*t*d(t) or 4*t*d(t)=1/8*d(w)

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

8. Substitute back w=1+16*t2 and add the constant of integration C

Final Answer

(∫_^)(arctan(4*t)*d(t))=t*arctan(4*t)−1/8*ln(1+16*t2)+C

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