Question 5 (10 points)
Use a trig. substitution to evaluate .

Ask by Brewer Colon. in the United States

Mar 09,2025

Question

Question 5 (10 points) Use a trig. substitution to evaluate $$ \int \frac{\sqrt{4-4 x^{2}}}{x^{2}} d x $$.

UpStudy AI · Accepted Answer

We begin with the integral

$$
\int \frac{\sqrt{4 - 4x^2}}{x^2} \, dx.
$$

**Step 1. Factor the square root.**

Notice that

$$
\sqrt{4-4x^2} = \sqrt{4(1-x^2)} = 2\sqrt{1-x^2}.
$$

Thus, the integral becomes

$$
\int \frac{2\sqrt{1-x^2}}{x^2} \, dx.
$$

**Step 2. Use the substitution $$ x = \sin \theta $$.**

With the substitution

$$
x = \sin \theta, \quad dx = \cos \theta \, d\theta,
$$

we have

$$
\sqrt{1-x^2} = \sqrt{1-\sin^2 \theta} = \cos \theta.
$$

Substitute these into the integral:

$$
\int \frac{2 \cos \theta}{\sin^2 \theta} \cdot \cos \theta \, d\theta = 2 \int \frac{\cos^2 \theta}{\sin^2 \theta} \, d\theta.
$$

**Step 3. Rewrite the integrand in terms of cotangent.**

Since

$$
\frac{\cos^2 \theta}{\sin^2 \theta} = \cot^2 \theta,
$$

the integral becomes

$$
2 \int \cot^2 \theta \, d\theta.
$$

Recall that

$$
\cot^2 \theta = \csc^2 \theta - 1.
$$

So the integral splits into

$$
2 \int (\csc^2 \theta - 1) \, d\theta = 2 \int \csc^2 \theta \, d\theta - 2 \int d\theta.
$$

**Step 4. Integrate term by term.**

We use the standard integrals

$$
\int \csc^2 \theta \, d\theta = -\cot \theta, \quad \int d\theta = \theta.
$$

Thus, we have

$$
2 \int \csc^2 \theta \, d\theta - 2 \int d\theta = 2(-\cot \theta) - 2 \theta = -2 \cot \theta - 2 \theta + C.
$$

**Step 5. Back-substitute in terms of $$ x $$.**

Recall that

$$
x = \sin \theta \quad \Longrightarrow \quad \theta = \arcsin x,
$$
$$
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{1-x^2}}{x}.
$$

Substituting these back gives

$$
-2 \cot \theta - 2 \theta + C = -2 \frac{\sqrt{1-x^2}}{x} - 2 \arcsin x + C.
$$

**Final Answer:**

$$
\boxed{-2\frac{\sqrt{1-x^2}}{x} - 2\arcsin x + C.}
$$
$$ \int \frac{\sqrt{4 - 4x^2}}{x^2} \, dx = -2\frac{\sqrt{1 - x^2}}{x} - 2\arcsin x + C $$

Question 5 (10 points)
Use a trig. substitution to evaluate .

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Question 5 (10 points) Use a trig. substitution to evaluate .