Question

$$ int e ^ { 2 x + 5 } d x $$

Explanation:

Step1: Use substitution

Let \(u = 2x+5\), then \(du=2dx\), and \(dx=\frac{1}{2}du\).

Step2: Substitute into the integral

\(\int e^{2x + 5}dx=\frac{1}{2}\int e^{u}du\)

Step3: Integrate \(e^{u}\)

Since \(\int e^{u}du=e^{u}+C\), we have \(\frac{1}{2}\int e^{u}du=\frac{1}{2}e^{u}+C\)

Step4: Substitute back \(u = 2x+5\)

\(\frac{1}{2}e^{u}+C=\frac{1}{2}e^{2x+5}+C\)

Answer:

\(\frac{1}{2}e^{2x + 5}+C\)