Evaluate the Integral

Problem

(∫_^)((x+1)/(x2+2*x)*d(x))

Solution

1. Identify the structure of the integrand and notice that the numerator is a scalar multiple of the derivative of the denominator.

2. Calculate the derivative of the denominator u=x2+2*x to verify the relationship.

d(u)/d(x)=2*x+2

3. Rewrite the derivative by factoring out a constant to match the numerator.

d(u)/d(x)=2*(x+1)

4. Substitute u=x2+2*x and d(u)=2*(x+1)*d(x) which implies (x+1)*d(x)=1/2*d(u)

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

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

1/2*(∫_^)(1/u*d(u))=1/2*ln(u)+C

6. Back-substitute the original expression for u to find the final result.

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

Final Answer

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

Want more problems? Check here!