https://file.aiquickdraw.com/mathbot/file/1726444970427qqggdysw.jpg

The image contains two parts: an integral and a right triangle with trigonometric references.

### The Integral:
\[
\int \frac{x^2}{\sqrt{1 - x^2}} \, dx
\]
This type of integral suggests using **trigonometric substitution** to simplify the expression.

### Trigonometric Substitution:
The right triangle on the right-hand side of the image provides a hint for the substitution. It shows the following:
- The adjacent side of the angle \(\theta\) is \(\sqrt{1 - x^2}\),
- The opposite side is \(x\),
- The hypotenuse is \(1\).

This setup corresponds to the trigonometric identity for sine:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{1} = x
\]
Thus, **the substitution being suggested is**:
\[
x = \sin(\theta)
\]
With this substitution:
- \( dx = \cos(\theta) d\theta \),
- \(\sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \cos(\theta)\).

Now, substitute into the integral:
\[
\int \frac{\sin^2(\theta)}{\cos(\theta)} \cdot \cos(\theta) \, d\theta
\]
The \(\cos(\theta)\) terms cancel out:
\[
\int \sin^2(\theta) \, d\theta
\]
This integral can be solved using the half-angle identity:
\[
\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}
\]
Thus, the integral becomes:
\[
\frac{1}{2} \int (1 - \cos(2\theta)) \, d\theta = \frac{1}{2} \left( \theta - \frac{\sin(2\theta)}{2} \right) + C
\]
Finally, substitute back using \( \theta = \arcsin(x) \), and use the appropriate identities for \(\sin(2\theta)\).

This shows that the triangle helps visualize the trigonometric substitution \( x = \sin(\theta) \), simplifying the integral step by step.

---

Would you like me to go into more detail on the final expression or any step above? Here are some related questions:

1. What happens if I substitute \( x = \cos(\theta) \) instead of \( \sin(\theta) \)?
2. How can I use a double-angle identity in integrals like this one?
3. What is the geometrical meaning of the right triangle in this substitution?
4. Can I solve this integral without using trigonometric substitution?
5. How does the half-angle identity simplify integrals involving powers of sine?

**Tip:** Trigonometric substitutions often simplify square root expressions when \(1 - x^2\) or similar terms are involved.

Why is trigonometric substitution used in this integral problem?

This tutorial explains how to solve the integral of x² over the square root of (1 - x²) using trigonometric substitution. Follow along with a step-by-step guide, using key trigonometric identities and substitution methods.

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