Evaluate the Integral

Problem

(∫_^)((x+3)/(x2+6*x+7)*d(x))

Solution

1. Identify the relationship between the numerator and the denominator. Notice that the derivative of the denominator x2+6*x+7 is 2*x+6

2. Rewrite the numerator to match a multiple of the derivative of the denominator. Since 2*x+6=2*(x+3) we can write the integral as:

(∫_^)((x+3)/(x2+6*x+7)*d(x))=1/2*(∫_^)((2*x+6)/(x2+6*x+7)*d(x))

3. Apply the substitution u=x2+6*x+7 which implies d(u)=(2*x+6)*d(x)

4. Substitute the variables into the integral:

1/2*(∫_^)(1/u*d(u))

5. Integrate using the natural logarithm rule (∫_^)(1/u*d(u))=ln(u)+C

1/2*ln(u)+C

6. Back-substitute the original expression for u

1/2*ln(x2+6*x+7)+C

Final Answer

(∫_^)((x+3)/(x2+6*x+7)*d(x))=1/2*ln(x2+6*x+7)+C

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