Evaluate the Integral

Problem

(∫_^)(x/(x2+2*x+2)*d(x))

Solution

1. Complete the square for the denominator to prepare for substitution.

x2+2*x+2=(x+1)2+1

2. Substitute u=x+1 to simplify the expression.

u=x+1

d(u)=d(x)

x=u−1

3. Rewrite the integral in terms of u

(∫_^)((u−1)/(u2+1)*d(u))

4. Split the integral into two separate terms.

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

5. Evaluate the first term using the substitution w=u2+1 where d(w)=2*u*d(u)

(∫_^)(u/(u2+1)*d(u))=1/2*ln(u2+1)

6. Evaluate the second term using the standard integral for the arctangent function.

(∫_^)(1/(u2+1)*d(u))=arctan(u)

7. Combine the results and add the constant of integration C

1/2*ln(u2+1)−arctan(u)+C

8. Back-substitute u=x+1 to return to the original variable x

1/2*ln((x+1)2+1)−arctan(x+1)+C

Final Answer

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

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