Integral Exercise 1

We are asked to evaluate the following integral:

$$ \int e^{5x} \ dx $$

To solve it, we’ll use the substitution method.

Let’s introduce a substitution to simplify the exponent:

$$ t = 5x $$

Differentiating both sides gives:

$$ dt = 5 \, dx \quad \Rightarrow \quad dx = \frac{1}{5} \, dt $$

We now rewrite the integral in terms of \( t \):

$$ \int e^t \, dx = \int e^t \cdot \frac{1}{5} \, dt $$

Factoring out the constant:

$$ \frac{1}{5} \int e^t \, dt $$

This is a standard integral, since:

\( \int e^t \, dt = e^t + C \)

So we obtain:

$$ \frac{1}{5} e^t + C $$

Substituting back \( t = 5x \), the final answer is:

$$ \frac{1}{5} e^{5x} + C $$

And that completes the solution.