Question

find the integral.
\\( \int 3 e ^ { 5 x } d x \\)
\\( \int 3 e ^ { 5 x } d x = \square \\)

Explanation:

Step1: Use the constant multiple rule

The constant multiple rule of integration states that \(\int kf(x)dx = k\int f(x)dx\) where \(k\) is a constant. For \(\int 3e^{5x}dx\), we can write it as \(3\int e^{5x}dx\).

Step2: Use substitution

Let \(u = 5x\), then \(du=5dx\), and \(dx=\frac{1}{5}du\). So \(3\int e^{5x}dx=3\int e^{u}\cdot\frac{1}{5}du\).

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

Since \(\int e^{u}du=e^{u}+C\), then \(3\int e^{u}\cdot\frac{1}{5}du=\frac{3}{5}\int e^{u}du\). Substituting \(\int e^{u}du = e^{u}+C\) gives \(\frac{3}{5}e^{u}+C\).

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

Substituting \(u = 5x\) back into \(\frac{3}{5}e^{u}+C\), we get \(\frac{3}{5}e^{5x}+C\).

Answer:

\(\frac{3}{5}e^{5x}+C\)