onsider the integral

:

This can be transformed into a basic integral by letting

After performing the substitution, you obtain the integral

It seems like your question involves a substitution method to transform an integral into a basic form. However, the details of the integral and substitution seem to be incomplete.

Let's walk through a general example of substitution. Consider an integral of the form:

\[
\int f(g(x)) g'(x) \, dx
\]

We typically perform a substitution by letting \( u = g(x) \). Then, \( du = g'(x) dx \), and the integral becomes:

\[
\int f(u) \, du
\]

Now, if you provide the specific form of the integral and the substitution, we can work through it.

Could you clarify the exact integral and substitution you'd like to perform?

Here are 5 possible substitution integrals to consider:

1. \( \int \sin(3x) \, dx \)
2. \( \int x e^{x^2} \, dx \)
3. \( \int \frac{1}{x^2 + 1} \, dx \)
4. \( \int \frac{1}{\sqrt{1 - x^2}} \, dx \)
5. \( \int \cos(2x) e^{\sin(2x)} \, dx \)

**Tip:** When choosing a substitution, try to simplify the integral by turning a complex expression into a basic form you can easily integrate.

Consider the integral. This can be transformed into a basic integral by letting. After performing the substitution, you obtain the integral.

Explore how to solve integrals using the substitution method. This example walks through transforming complex integrals into basic forms by applying u-substitution. Suitable for advanced high school students and college-level calculus learners.

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