Question

$\int x^4 e^{x^5} \cos\left(e^{x^5}\right)\, dx$ is equal to:

• A. $\frac{1}{3} \sin(e^{x^5}) + C$
• B. $\frac{1}{4} \sin(e^{x^5}) + C$
• C. $\frac{1}{5} \sin(e^{x^5}) + C$
• D. $\sin(e^{x^5}) + C$
• E. $2 \sin(e^{x^5}) + C$

Hint:

When multiple functions are nested, start from the innermost part. Often, the exponential function's derivative includes both the exponential itself and the power function's derivative, making it a perfect candidate for $u$-substitution.

Answer 1

The Correct Option is C

Concept: This problem requires integration by substitution. Look for a complicated nested function whose derivative (or a multiple of it) appears elsewhere in the integrand. Here, $e^{x^5}$ is the nested function.

Step 1: Apply the substitution.
Let $u = e^{x^5}$. Differentiate with respect to $x$ using the chain rule: \[ \frac{du}{dx} = e^{x^5} \cdot \frac{d}{dx}(x^5) = e^{x^5} \cdot 5x^4 \] \[ du = 5x^4 e^{x^5} dx \quad \Rightarrow \quad \frac{1}{5} du = x^4 e^{x^5} dx \]

Step 2: Rewrite the integral in terms of $u$.
Substitute the terms identified in Step 1 into the original integral: \[ \int \cos(u) \cdot \frac{1}{5} du \] \[ = \frac{1}{5} \int \cos(u) du \]

Step 3: Integrate and substitute back.
Since $\int \cos(u) du = \sin(u) + C$: \[ = \frac{1}{5} \sin(u) + C \] Substitute back $u = e^{x^5}$: \[ = \frac{1}{5} \sin(e^{x^5}) + C \]