Definite Integration
Integration
Grade Class 12
Question:
$\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$ equals
$C - \ln(1 + (x + 1)e^x) - \frac{1}{1 + (x + 1)e^x}$
$C - \ln(1 + (x + 1)e^x) + \frac{1}{1 + (x - 1)e^x}$
$C - \ln(1 - (x^2 - 1)e^{-x}) + \frac{1}{1 + (x + 1)e^x}$
$C - \ln(1 + (x + 1)e^{-x}) - \frac{1}{1 + (x + 1)e^{-x}}$
Step-by-Step Solution
Key Concept: The integral can be solved by manipulating the integrand to a form suitable for substitution or by recognizing it as a derivative of a quotient involving exponential functions.
The integral is $\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$. By rewriting the integrand and using substitution, we can simplify the expression to a standard form. The correct option is (C).
Correct Answer: 3