Integral Exercise 15
We are tasked with evaluating the following integral:
$$ \int 2x \cdot e^{x^2} \ dx $$
To solve it, we’ll apply the method of substitution.
Let’s introduce a substitution to simplify the integrand:
$$ u = x^2 $$
Now, differentiate both sides:
$$ du = 2x \ dx $$
We can now rewrite the integral in terms of \( u \):
$$ \int 2x \cdot e^{x^2} \ dx $$
Substituting \( u = x^2 \) and \( 2x \, dx = du \), we get:
$$ \int e^u \ du $$
This is a basic exponential integral, whose antiderivative is well known:
∫ eu du = eu + c
Therefore:
$$ \int e^u \ du = e^u + c $$
Now, substituting back \( u = x^2 \), we find the solution to the original integral:
$$ e^{x^2} + c $$
So, the final result is:
$$ \int 2x \cdot e^{x^2} \ dx = e^{x^2} + c $$
And that's the complete solution.
