Question

Evaluate the following integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \]

• A. \( \frac{e^x}{1+x(1+x)^2} \)
• B. \( \int \frac{e^x}{(1+x(1+x)^2)} dx \)
• C. \( \frac{e^x}{(1+x(1+x))^2} \)
• D. None of the above

Hint:

For integrals involving exponential functions and rational expressions, try simplifying the rational part to see if any standard integrals apply.

Answer 1

The Correct Option is A

We are asked to evaluate the integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \] First, simplify the function inside the integral. Begin by expanding \( 1 + x(1 + x)^2 \): \[ 1 + x(1 + x)^2 = 1 + x(1 + 2x + x^2) = 1 + x + 2x^2 + x^3 \] The integral becomes: \[ \int e^x \left[\frac{1}{1+x + 2x^2 + x^3}\right] dx \] Recognizing the structure of the integral, this can be simplified to: \[ \frac{e^x}{1 + x + 2x^2 + x^3} \] Thus, the correct answer is: \[ \boxed{(A) \frac{e^x}{1+x(1+x)^2}} \]