Calculus Examples

$\int \frac{1}{\sqrt{4 - 9 x^{2}}} d x$

Step 1

Write the expression using exponents.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\int \frac{1}{\sqrt{2^{2} - {(3 x)}^{2}}} d x$

Step 2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 2.1.3

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

$3 \cdot 1$

Step 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

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

$\int \frac{1}{\sqrt{2^{2} - u^{2}}} \cdot \frac{1}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{\sqrt{2^{2} - u^{2}}} d u$

Step 4

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

$\frac{1}{3} (\arcsin (\frac{u}{2}) + C)$

Step 5

Rewrite $\frac{1}{3} (\arcsin (\frac{u}{2}) + C)$ as $\frac{1}{3} \arcsin (\frac{1}{2} u) + C$ .

$\frac{1}{3} \arcsin (\frac{1}{2} u) + C$

Step 6

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{3} \arcsin (\frac{1}{2} \cdot 3 x) + C$