nswer The evaluated integral is: \[ \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \left(\frac{x}{a}\right) + C \]2d:T7ac,To evaluate the integral \( I = \int \frac{dx}{\sqrt{a^2 - x^2}} \), we can use a trigonometric substitution. Here’s a step-by-step solution:
### Step 1: Trigonometric Substitution
Let \( x = a \sin \theta \). Then, the differential \( dx \) becomes:
\[
dx = a \cos \theta \, d\theta
\]
**Hint:** Use the substitution \( x = a \sin \theta \) to simplify the integral involving a square root.
### Step 2: Substitute in the Integral
Substituting \( x \) and \( dx \) into the integral gives:
\[
I = \int \frac{a \cos \theta \, d\theta}{\sqrt{a^2 - (a \sin \theta)^2}}
\]
### Step 3: Simplify the Denominator
Now simplify the expression under the square root:
\[
a^2 - (a \sin \theta)^2 = a^2 (1 - \sin^2 \theta) = a^2 \cos^2 \theta
\]
Thus, the square root becomes:
\[
\sqrt{a^2 \cos^2 \theta} = a \cos \theta
\]
**Hint:** Remember that \( \sqrt{a^2 \cos^2 \theta} = a |\cos \theta| \). Since \( \theta \) is in the range where \( \cos \theta \) is positive, we can write it as \( a \cos \theta \).
### Step 4: Substitute Back into the Integral
Now, substitute this back into the integral:
\[
I = \int \frac{a \cos \theta \, d\theta}{a \cos \theta}
\]
This simplifies to:
\[
I = \int d\theta
\]
**Hint:** After simplification, the integral reduces to a basic integral of \( d\theta \).
### Step 5: Integrate
The integral of \( d\theta \) is:
\[
I = \theta + C
\]
### Step 6: Back Substitute for \( \theta \)
Recall our substitution \( x = a \sin \theta \), which implies:
\[
\sin \theta = \frac{x}{a} \quad \Rightarrow \quad \theta = \sin^{-1} \left(\frac{x}{a}\right)
\]
Thus, we can write:
\[
I = \sin^{-1} \left(\frac{x}{a}\right) + C
\]
### Final Answer
The evaluated integral is:
\[
\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}

Evaluate `intdx/sqrt (a^2-x^2)`

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