Evaluate the Integral

Problem

(∫_^)(x*arctan(x2)*d(x))

Solution

1. Substitute a new variable to simplify the integrand. Let u=x2 Then d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

2. Rewrite the integral in terms of u

(∫_^)(x*arctan(x2)*d(x))=1/2*(∫_^)(arctan(u)*d(u))

3. Apply integration by parts to the integral (∫_^)(arctan(u)*d(u)) Let w=arctan(u) and d(v)=d(u) Then d(w)=1/(1+u2)*d(u) and v=u

4. Use the formula (∫_^)(w*d(v))=w*v−(∫_^)(v*d(w))

(∫_^)(arctan(u)*d(u))=u*arctan(u)−(∫_^)(u/(1+u2)*d(u))

5. Evaluate the remaining integral using the substitution s=1+u2 where d(s)=2*u*d(u)

(∫_^)(u/(1+u2)*d(u))=1/2*ln(1+u2)

6. Combine the results and include the constant factor of 1/2 from step 2.

1/2*(u*arctan(u)−1/2*ln(1+u2))+C

7. Back-substitute u=x2 to return to the original variable x

1/2*x2*arctan(x2)−1/4*ln(1+x4)+C

Final Answer

(∫_^)(x*arctan(x2)*d(x))=1/2*x2*arctan(x2)−1/4*ln(1+x4)+C

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