egral gives: \[ I = -\frac{1}{a} \cdot \frac{\sqrt{a^2 - x^2}}{x} + C. \] Thus, the final answer is: \[ I = -\frac{\sqrt{a^2 - x^2}}{ax} + C. \]2d:T7d5,To evaluate the integral
\[
I = \int \frac{dx}{x^2 \sqrt{a^2 - x^2}},
\]
we will use the substitution \( x = a \sin \theta \).
### Step 1: Substitution
Let \( x = a \sin \theta \). Then, we differentiate both sides to find \( dx \):
\[
dx = a \cos \theta \, d\theta.
\]
### Step 2: Substitute in the Integral
Now, we substitute \( x \) and \( dx \) into the integral:
\[
I = \int \frac{a \cos \theta \, d\theta}{(a \sin \theta)^2 \sqrt{a^2 - (a \sin \theta)^2}}.
\]
### Step 3: Simplify the Integral
We simplify the expression inside the integral:
\[
I = \int \frac{a \cos \theta \, d\theta}{a^2 \sin^2 \theta \sqrt{a^2(1 - \sin^2 \theta)}}.
\]
Since \( \sqrt{a^2(1 - \sin^2 \theta)} = a \cos \theta \), we can rewrite the integral as:
\[
I = \int \frac{a \cos \theta \, d\theta}{a^2 \sin^2 \theta \cdot a \cos \theta} = \int \frac{d\theta}{a \sin^2 \theta}.
\]
### Step 4: Further Simplification
This simplifies to:
\[
I = \frac{1}{a} \int \csc^2 \theta \, d\theta.
\]
### Step 5: Integrate
The integral of \( \csc^2 \theta \) is:
\[
\int \csc^2 \theta \, d\theta = -\cot \theta + C.
\]
Thus, we have:
\[
I = -\frac{1}{a} \cot \theta + C.
\]
### Step 6: Back Substitute
Now we need to back substitute for \( \theta \). Recall that \( x = a \sin \theta \), so:
\[
\sin \theta = \frac{x}{a} \implies \theta = \sin^{-1} \left( \frac{x}{a} \right).
\]
Using the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and knowing \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \frac{x^2}{a^2}} = \frac{\sqrt{a^2 - x^2}}{a} \), we find:
\[
\cot \theta = \frac{\sqrt{a^2 - x^2}}{x}.
\]
### Final Result
Substituting back into our integral gives:
\[
I = -\frac{1}{a} \cdot \frac{\sqrt{a

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

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