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

Problem

(∫_^)(arctan(2*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(2*t) and d(v)=d(t)

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

4. Substitute these into the integration by parts formula to get t*arctan(2*t)−(∫_^)((2*t)/(1+4*t2)*d(t))

5. Evaluate the remaining integral using usubstitution, letting w=1+4*t2 which implies d(w)=8*t*d(t) or 2*t*d(t)=1/4*d(w)

6. Integrate the substitution term 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*t)*d(t))=t*arctan(2*t)−1/4*ln(1+4*t2)+C

Want more problems? Check here!