find the integeral of (5x^2)/(sqrt(1-x^2) using trig subsitution

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.

find the integral of (5x^2)/(sqrt(1-x^2)) using trig substitution

Learn how to solve the integral of (5x^2)/(sqrt(1-x^2)) step-by-step using trigonometric substitution. This solution involves key calculus concepts like trigonometric identities and integral techniques, suitable for undergraduate calculus students.

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