Question

\( \int \frac{(1 + x) e^x{\sin^2(x e^x)} dx \) is equal to:}

• A. \( -\cot(e^x) + C \)
• B. \( \tan(x e^x) + C \)
• C. \( \tan(e^x) + C \)
• D. \( \cot(x e^x) + C \)
• E. \( -\cot(x e^x) + C \)

Hint:

If you see $e^x$ and $(1+x)$ in an integral, always check if $x e^x$ is your $u$! It's one of the most common substitution patterns in calculus.

Answer 1

Answer: (E)

Concept: The integral can be simplified by substituting the argument of the sine function. We recognize that the numerator is the derivative of the argument.

Step 1: Apply substitution.
Let \( u = x e^x \). Using the product rule for differentiation: \[ du = (1 \cdot e^x + x \cdot e^x) dx = e^x(1 + x) dx \]

Step 2: Substitute and integrate.
The integral becomes: \[ \int \frac{du}{\sin^2 u} = \int \csc^2 u \, du \]

Step 3: Evaluate and substitute back.
The integral of \( \csc^2 u \) is \( -\cot u \). \[ -\cot(x e^x) + C \]