To solve the given integral:

$\int \frac{x^2}{\sqrt{441 - x^2}} \, dx$

we will use trigonometric substitution to simplify it.

Part 1: Choosing the appropriate substitution

We are given three options:

• $x = 21 \sin \theta$
• $x = 21 \sec \theta$
• $x = 21 \tan \theta$

Since the integrand involves a square root of the form $\sqrt{a^2 - x^2}$ (where $a = 21$), the appropriate substitution is:

$x = 21 \sin \theta$

This substitution is correct because it simplifies the radical $\sqrt{441 - x^2}$ by using the identity $1 - \sin^2 \theta = \cos^2 \theta$.

Part 2: Rewrite the integral using the substitution

Given $x = 21 \sin \theta$, we can differentiate to find:

$dx = 21 \cos \theta \, d\theta$

Now substitute $x = 21 \sin \theta$ into the integral:

1. The expression $\sqrt{441 - x^2}$ becomes: $\sqrt{441 - x^2} = \sqrt{441 - 441 \sin^2 \theta} = \sqrt{441 \cos^2 \theta} = 21 \cos \theta$

2. The expression $x^2$ becomes: $x^2 = (21 \sin \theta)^2 = 441 \sin^2 \theta$

Thus, the integral transforms into:

$\int \frac{441 \sin^2 \theta}{21 \cos \theta} \cdot 21 \cos \theta \, d\theta$

Simplifying:

$\int 441 \sin^2 \theta \, d\theta$

Part 3: Evaluating the integral

To evaluate $\int 441 \sin^2 \theta \, d\theta$, we use the identity:

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

Substituting this into the integral:

$\int 441 \sin^2 \theta \, d\theta = 441 \int \frac{1 - \cos(2\theta)}{2} \, d\theta$

This simplifies to:

$\frac{441}{2} \int (1 - \cos(2\theta)) \, d\theta$

Now integrate each term:

$\frac{441}{2} \left( \theta - \frac{\sin(2\theta)}{2} \right) + C$

Step 4: Back-substitution

Since $x = 21 \sin \theta$, we know:

$\sin \theta = \frac{x}{21}$

Thus, $\theta = \arcsin\left(\frac{x}{21}\right)$.

Now substitute back in terms of $x$:

$\frac{441}{2} \left( \arcsin\left(\frac{x}{21}\right) - \frac{1}{2} \sin\left(2\arcsin\left(\frac{x}{21}\right)\right) \right) + C$

Using the identity $\sin(2\theta) = 2\sin\theta \cos\theta$, and knowing that $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \frac{x^2}{441}}$, we can simplify further. However, this expression represents the final answer in terms of $x$.

Final Answer:

$\frac{441}{2} \left( \arcsin\left(\frac{x}{21}\right) - \frac{x\sqrt{441 - x^2}}{441} \right) + C$

Would you like a deeper explanation of any step?

Here are five follow-up questions for further practice:

1. What trigonometric substitution would you use for an integral of the form $\int \frac{x^2}{\sqrt{x^2 - a^2}} \, dx$?
2. How would the integral change if the upper limit was $21$?
3. Can you apply this method to an integral involving $\sqrt{x^2 + a^2}$?
4. How does trigonometric substitution simplify radicals like $\sqrt{a^2 - x^2}$?
5. What are the geometric interpretations of the arcsine and cosine functions in this context?

Tip: When performing trigonometric substitutions, always check the form of the radical to pick the right trigonometric identity.