Evaluate the Integral

Problem

(∫_0^1)(arctan(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: let u=arctan(x) and d(v)=d(x)

3. Differentiate u to find d(u)=1/(1+x2)*d(x) and integrate d(v) to find v=x

4. Apply the integration by parts formula to the definite integral:

(∫_0^1)(arctan(x)*d(x))=[x*arctan(x)]10−(∫_0^1)(x/(1+x2)*d(x))

5. Evaluate the boundary term [x*arctan(x)]10

1⋅arctan(1)−0⋅arctan(0)=π/4

6. Solve the remaining integral (∫_0^1)(x/(1+x2)*d(x)) using the substitution w=1+x2 which gives d(w)=2*x*d(x)

(∫_0^1)(x/(1+x2)*d(x))=1/2*[ln(1+x2)]10

7. Evaluate the result of the substitution:

1/2*(ln(2)−ln(1))=1/2*ln(2)

8. Combine the results to find the final value:

π/4−1/2*ln(2)

Final Answer

(∫_0^1)(arctan(x)*d(x))=π/4−ln(2)/2

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