Question

evaluate. be sure to check by differentiating
int e ^ { 4 x + 2 } d x
int e ^ { 4 x + 2 } d x = square
(type an exact answer. use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Use substitution

Let \(u = 4x+2\), then \(du=4dx\), and \(dx=\frac{1}{4}du\).
The integral \(\int e^{4x + 2}dx\) becomes \(\frac{1}{4}\int e^{u}du\).

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

Since \(\int e^{u}du=e^{u}+C\), then \(\frac{1}{4}\int e^{u}du=\frac{1}{4}e^{u}+C\).

Step3: Substitute back \(u = 4x + 2\)

We get \(\frac{1}{4}e^{4x+2}+C\).

Step4: Check by differentiating

Differentiate \(y=\frac{1}{4}e^{4x + 2}+C\) using the chain - rule.
If \(y=\frac{1}{4}e^{4x + 2}+C\), then \(y^\prime=\frac{1}{4}\times e^{4x + 2}\times4=e^{4x+2}\).

Answer:

\(\frac{1}{4}e^{(4x + 2)}+C\)