Evaluate the following integral using trigonometric substitution.
Integral from nothing to nothing StartFraction x squared Over StartRoot 441 minus x squared EndRoot EndFraction dx
Question content area bottom
Part 1
What substitution will be the most helpful for evaluating this​ integral?
A.
xequals21 sine theta
Your answer is correct.B.
xequals21 secant theta
C.
xequals21 tangent theta
Part 2
Rewrite the given integral using this substitution.  
Integral from nothing to nothing StartFraction x squared Over StartRoot 441 minus x squared EndRoot EndFraction dx
equalsIntegral from nothing to nothing left parenthesis 441 left parenthesis StartFraction 1 minus cosine left parenthesis 2 theta right parenthesis Over 2 EndFraction right parenthesis right parenthesis d theta
  
​(Type an exact​ answer.)
Part 3
Evaluate the integral.

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.

Evaluate the following integral using trigonometric substitution.
Integral from nothing to nothing StartFraction x squared Over StartRoot 441 minus x squared EndRoot EndFraction dx
Question content area bottom
Part 1
What substitution will be the most helpful for evaluating this​ integral?
A.
x equals 21 sine theta
B.
x equals 21 secant theta
C.
x equals 21 tangent theta
Part 2
Rewrite the given integral using this substitution. Integral from nothing to nothing StartFraction x squared Over StartRoot 441 minus x squared EndRoot EndFraction dx equals Integral from nothing to nothing left parenthesis 441 left parenthesis StartFraction 1 minus cosine left parenthesis 2 theta right parenthesis Over 2 EndFraction right parenthesis right parenthesis d theta
(Type an exact​ answer.)
Part 3
Evaluate the integral.

Learn how to evaluate the integral of x^2 / √(441 - x^2) using trigonometric substitution. This problem involves key calculus concepts, including trigonometric identities and integration techniques. Suitable for undergraduate calculus students.

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