Evaluate the Integral integral of arctan(8t) with respect to t

Problem

(∫_^)(arctan(8*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(8*t) and d(v)=d(t)

3. Differentiate u to find d(u) using the chain rule, resulting in d(u)=8/(1+(8*t)2)*d(t)=8/(1+64*t2)*d(t)

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

5. Substitute these into the integration by parts formula:

(∫_^)(arctan(8*t)*d(t))=t*arctan(8*t)−(∫_^)((8*t)/(1+64*t2)*d(t))

6. Evaluate the remaining integral using usubstitution, letting w=1+64*t2 which implies d(w)=128*t*d(t) or 8*t*d(t)=1/16*d(w)

7. Substitute and integrate:

(∫_^)((8*t)/(1+64*t2)*d(t))=(∫_^)(1/(16*w)*d(w))

(∫_^)(1/(16*w)*d(w))=1/16*ln(w)

8. Back-substitute w=1+64*t2 and combine all terms, adding the constant of integration C

Final Answer

(∫_^)(arctan(8*t)*d(t))=t*arctan(8*t)−1/16*ln(1+64*t2)+C

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