To solve the integral $\int \frac{5x^2}{\sqrt{1 - x^2}} \, dx$ using trigonometric substitution, we follow these steps:

Step 1: Trigonometric Substitution

We will use the substitution:

$x = \sin \theta \quad \text{so that} \quad dx = \cos \theta \, d\theta$ Also, note that: $\sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta$

Step 2: Rewrite the Integral

Substitute $x = \sin \theta$ and $dx = \cos \theta \, d\theta$ into the integral:

$\int \frac{5x^2}{\sqrt{1 - x^2}} \, dx = \int \frac{5 \sin^2 \theta}{\cos \theta} \cdot \cos \theta \, d\theta$

Since $\cos \theta$ cancels out, the integral simplifies to:

$\int 5 \sin^2 \theta \, d\theta$

Step 3: Simplify Using a Trigonometric Identity

We use the identity for $\sin^2 \theta$:

$\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$

Substitute this into the integral:

$\int 5 \sin^2 \theta \, d\theta = \int 5 \cdot \frac{1 - \cos(2\theta)}{2} \, d\theta$

Simplify:

$\int 5 \sin^2 \theta \, d\theta = \frac{5}{2} \int (1 - \cos(2\theta)) \, d\theta$

Step 4: Integrate

Now, integrate each term:

$\frac{5}{2} \left( \int 1 \, d\theta - \int \cos(2\theta) \, d\theta \right)$

The integrals are:

$\int 1 \, d\theta = \theta$ $\int \cos(2\theta) \, d\theta = \frac{\sin(2\theta)}{2}$

So the result is:

$\frac{5}{2} \left( \theta - \frac{\sin(2\theta)}{2} \right) = \frac{5}{2} \theta - \frac{5}{4} \sin(2\theta)$

Step 5: Back-Substitute

Recall the original substitution $x = \sin \theta$, so $\theta = \arcsin(x)$. Also, $\sin(2\theta) = 2\sin\theta \cos\theta = 2x \sqrt{1 - x^2}$.

Thus, the integral becomes:

$\frac{5}{2} \arcsin(x) - \frac{5}{4} \cdot 2x \sqrt{1 - x^2}$

Simplifying:

$\frac{5}{2} \arcsin(x) - \frac{5x \sqrt{1 - x^2}}{2}$

Final Answer:

The integral is:

$\int \frac{5x^2}{\sqrt{1 - x^2}} \, dx = \frac{5}{2} \arcsin(x) - \frac{5x \sqrt{1 - x^2}}{2} + C$

Do you have any questions or would you like further details?

Here are 5 related questions:

1. How do you apply other substitutions in integration?
2. Can you integrate functions involving $\cos^2(x)$ using trigonometric identities?
3. How do you solve integrals involving exponential functions using substitution?
4. What are the benefits of using trigonometric substitution in integration?
5. Can you explain how to solve an integral involving $\tan(x)$ using substitution?

Tip: Always check for trigonometric identities or algebraic simplifications before integrating directly.