AskSia

Plus

\int \frac{d x}{x^{2} \sqrt{9-x^{2}}}

Jan 29, 2024

\int \frac{d x}{x^{2} \sqrt{9-x^{2}}}

Solution by Steps

step 1

To integrate the function

\frac{1}{x^{2} \sqrt{9-x^{2}}}

, we can use trigonometric substitution

step 2

Let

x = 3\sin(\theta)

, then

dx = 3\cos(\theta)d\theta

and

\sqrt{9-x^{2}} = \sqrt{9-9\sin^{2}(\theta)} = 3\cos(\theta)

step 3

Substituting

x

and

dx

into the integral, we get

\int \frac{3\cos(\theta)d\theta}{(3\sin(\theta))^{2} \cdot 3\cos(\theta)}

step 4

Simplify the integral to

\int \frac{d\theta}{9\sin^{2}(\theta)}

step 5

This integral is equivalent to

\int \frac{d\theta}{(3\sin(\theta))^{2}}

, which is

\int \csc^{2}(\theta)d\theta

step 6

The integral of

\csc^{2}(\theta)

-\cot(\theta)

step 7

Substituting back for

x

, we have

-\cot(\theta) = -\frac{\sqrt{9-x^{2}}}{x}

step 8

Therefore, the integral

\int \frac{1}{x^{2} \sqrt{9-x^{2}}} dx

-\frac{\sqrt{9-x^{2}}}{9x} + C

, where

C

is the constant of integration

Answer

-\frac{\sqrt{9-x^{2}}}{9x} + C

Key Concept

Trigonometric Substitution in Integration

Explanation

The integral of a function involving a square root of a quadratic can often be solved by trigonometric substitution, which simplifies the integral into a form that can be easily integrated.