QuestionEvaluate the integral: $\int \frac{1}{\sqrt{1-4 x^{2}}} d x$ .

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{1}{\sqrt{1-4 x^{}}} d x$ . We are going to use the integral formula $\int \frac{1}{\sqrt{a^{}-x^{}}} d x = \sin^{-1}\left(\frac{x}{a}\right) + C$ where $a$ is a constant and $C$ is the constant of integration.

STEP 2

First, we need to make the integral look like the formula. We can do this by factoring out the constant from under the square root.
$\int \frac{1}{\sqrt{1-4 x^{2}}} d x = \int \frac{1}{\sqrt{(2^2)-(2x)^2}} d x$

STEP 3

Now, we can see that our integral matches the formula with $a =2$ and $x$ replaced by $2x$ . So, we can apply the formula to get the antiderivative.
$\int \frac{1}{\sqrt{1- x^{2}}} d x = \sin^{-1}\left(\frac{2x}{2}\right) + C$

STEP 4

implify the fraction inside the arcsin function.
$\int \frac{1}{\sqrt{1-4 x^{2}}} d x = \sin^{-1}(x) + C$ So, the integral of $\frac{1}{\sqrt{1-4 x^{2}}}$ with respect to $x$ is $\sin^{-1}(x) + C$ .

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