Calculus Examples

$\int \frac{1}{\sqrt{1 - 4 x^{2}}} d x$

Step 1

Write the expression using exponents.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{\sqrt{1^{2} - 4 x^{2}}} d x$

Step 1.2

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\int \frac{1}{\sqrt{1^{2} - {(2 x)}^{2}}} d x$

Step 2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{\sqrt{1^{2} - u^{2}}} \cdot \frac{1}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{1}{\sqrt{1^{2} - u^{2}}} d u$

Step 4

The integral of $\frac{1}{\sqrt{1^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (u)$

$\frac{1}{2} (\arcsin (u) + C)$

Step 5

Simplify.

$\frac{1}{2} \arcsin (u) + C$

Step 6

Replace all occurrences of $u$ with $2 x$ .

$\frac{1}{2} \arcsin (2 x) + C$