https://file.aiquickdraw.com/mathbot/file/1726447386606glyuh0v5.jpg

why does the 1 goes there why does the x goes there and the operation below with the roof and also what if it was x^2-1 instead ?why do they went there is there a rule?

The image you've uploaded shows an integral and a corresponding right triangle used to apply a trigonometric substitution, which helps in solving the integral.

The integral is:

\[
\int \frac{x^2}{\sqrt{1 - x^2}} \, dx
\]

To understand the triangle and why the sides are labeled the way they are, we can break this down step by step, starting with trigonometric substitution.

### Why do the sides go where they do?
This is based on **trigonometric substitution**, which is used to simplify integrals involving square roots of quadratic expressions like \(1 - x^2\). In this case, a common substitution is:

\[
x = \sin(\theta)
\]

This substitution works because of the Pythagorean identity:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

When \(x = \sin(\theta)\), we know that \(1 - x^2 = \cos^2(\theta)\). Here's how this relates to the triangle.

1. **The hypotenuse** of the triangle is 1 because the Pythagorean identity is structured as \(1 - x^2\).
   
2. **The opposite side** is labeled \(x\), because \(x = \sin(\theta)\). In a right triangle, \(\sin(\theta)\) is defined as the ratio of the opposite side to the hypotenuse. Since the hypotenuse is 1, the opposite side is simply \(x\).
   
3. **The adjacent side** is \(\sqrt{1 - x^2}\), because the Pythagorean theorem tells us that the adjacent side should be \(\sqrt{1^2 - x^2} = \sqrt{1 - x^2}\).

Thus, the triangle helps visualize the substitution \(x = \sin(\theta)\), where:
- Hypotenuse \(= 1\)
- Opposite \(= x\)
- Adjacent \(= \sqrt{1 - x^2}\)

### What if it were \(x^2 - 1\) instead?
If the expression under the square root was \(x^2 - 1\) instead of \(1 - x^2\), we would use a different trigonometric substitution. In this case, we use:

\[
x = \sec(\theta)
\]

This works because:

\[
\sec^2(\theta) - 1 = \tan^2(\theta)
\]

So, the corresponding triangle for this case would be structured as:
- Hypotenuse \(= x\) (since \(\sec(\theta) = \frac{1}{\cos(\theta)}\))
- Adjacent \(= 1\)
- Opposite \(= \sqrt{x^2 - 1}\) (since \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\))

### Rule Behind the Substitution:
The trigonometric substitutions are based on the Pythagorean identities:
1. \(1 - x^2\): Use \(x = \sin(\theta)\) because \(1 - \sin^2(\theta) = \cos^2(\theta)\).
2. \(x^2 - 1\): Use \(x = \sec(\theta)\) because \(\sec^2(\theta) - 1 = \tan^2(\theta)\).
3. \(1 + x^2\): Use \(x = \tan(\theta)\) because \(1 + \tan^2(\theta) = \sec^2(\theta)\).

These substitutions help simplify square roots and make integrals easier to solve using trigonometric identities.

### Summary
- The side labeled \(x\) corresponds to the trigonometric substitution \(x = \sin(\theta)\).
- The side labeled \(\sqrt{1 - x^2}\) comes from the Pythagorean identity \(1 - \sin^2(\theta) = \cos^2(\theta)\).
- If the expression were \(x^2 - 1\), you would use the substitution \(x = \sec(\theta)\), and the triangle would be different.

Would you like a more detailed step-by-step solution to the integral, or do you have more questions about this? 

Here are 5 related questions:
1. How do you choose the right trigonometric substitution for an integral?
2. What other trigonometric identities are commonly used in calculus?
3. What happens when you use substitution in this specific integral \(\int \frac{x^2}{\sqrt{1 - x^2}} \, dx\)?
4. How do you evaluate the new integral after the substitution?
5. Can trigonometric substitution be used in non-square root integrals?

**Tip**: Always check if an integral involves a square root of a quadratic expression. Trigonometric substitution can often simplify these.

Why does the 1 go there, why does the x go there, and why does the operation below with the square root behave that way? Also, what if it was x^2 - 1 instead? Why did they go there? Is there a rule?

Explore the process of trigonometric substitution to solve integrals involving square roots, such as √(1 - x²), using the substitution x = sin(θ). Learn why the triangle sides are labeled as they are and how to handle expressions like x² - 1.

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