Question

find the indefinite integral and check the result by differentiating.

$$ int e ^ { 4 x } ( 4 ) d x $$

$$ int e ^ { 4 x } ( 4 ) d x = $$

Explanation:

Step1: Use substitution method

Let \( u = 4x \), then \( du=4dx \).
The integral \( \int e^{4x}(4)dx=\int e^{u}du \).

Step2: Integrate \( e^{u} \)

Since \( \int e^{u}du=e^{u}+C \) (where \( C \) is the constant of integration).
Substituting back \( u = 4x \), we get \( e^{4x}+C \).

Step3: Check by differentiating

Differentiate \( y = e^{4x}+C \) using the chain rule.
If \( y = e^{4x}+C \), then \( y^\prime=\frac{d}{dx}(e^{4x})+\frac{d}{dx}(C) \).
By the chain rule \( \frac{d}{dx}(e^{4x})=e^{4x}\cdot\frac{d}{dx}(4x)=4e^{4x} \), and \( \frac{d}{dx}(C) = 0 \).

Answer:

\( e^{4x}+C \)