Definite Integration
Integration
Grade Class 12
Question:
$\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$ equals
(A) C - \ln(1+(x+1)e^x) - \frac{1}{1+(x+1)e^x}
(B) C - \ln(1+(x+1)e^x) + \frac{1}{1-(x-1)e^x}
(C) C - \ln(1-(x^2-1)e^{-x}) + \frac{1}{1+(x+1)e^x}
(D) C - \ln(1+(x+1)e^{-x}) - \frac{1}{1+(x+1)e^{-x}}
Step-by-Step Solution
Key Concept: Manipulate the integrand to identify a derivative form.
The integral can be solved by rearranging the terms and using substitution.
Correct Answer: D