Calculus Examples

$\int \frac{- 4}{\sqrt{16 - 36 x^{2}}} d x$

Step 1

Move the negative in front of the fraction.

$\int - \frac{4}{\sqrt{16 - 36 x^{2}}} d x$

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{4}{\sqrt{16 - 36 x^{2}}} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- (4 \int \frac{1}{\sqrt{16 - 36 x^{2}}} d x)$

Step 4

Write the expression using exponents.

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Step 4.1

Multiply $4$ by $- 1$ .

$- 4 \int \frac{1}{\sqrt{16 - 36 x^{2}}} d x$

Step 4.2

Rewrite $16$ as $4^{2}$ .

$- 4 \int \frac{1}{\sqrt{4^{2} - 36 x^{2}}} d x$

Step 4.3

Rewrite $36 x^{2}$ as ${(6 x)}^{2}$ .

$- 4 \int \frac{1}{\sqrt{4^{2} - {(6 x)}^{2}}} d x$

Step 5

Let $u = 6 x$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 6 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 5.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 5.1.3

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

$6 \cdot 1$

Step 5.1.4

Multiply $6$ by $1$ .

$6$

Step 5.2

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

$- 4 \int \frac{1}{\sqrt{4^{2} - u^{2}}} \cdot \frac{1}{6} d u$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$- 4 (\frac{1}{6} \int \frac{1}{\sqrt{4^{2} - u^{2}}} d u)$

Step 7

The integral of $\frac{1}{\sqrt{4^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (\frac{u}{4})$

$- 4 (\frac{1}{6} (\arcsin (\frac{u}{4}) + C))$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{6}$ and $- 4$ .

$\frac{- 4}{6} (\arcsin (\frac{u}{4}) + C)$

Step 8.1.2

Cancel the common factor of $- 4$ and $6$ .

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Step 8.1.2.1

Factor $2$ out of $- 4$ .

$\frac{2 (- 2)}{6} (\arcsin (\frac{u}{4}) + C)$

Step 8.1.2.2

Cancel the common factors.

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Step 8.1.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot - 2}{2 \cdot 3} (\arcsin (\frac{u}{4}) + C)$

Step 8.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 \cdot 3} (\arcsin (\frac{u}{4}) + C)$

Step 8.1.2.2.3

Rewrite the expression.

$\frac{- 2}{3} (\arcsin (\frac{u}{4}) + C)$

Step 8.1.3

Move the negative in front of the fraction.

$- \frac{2}{3} (\arcsin (\frac{u}{4}) + C)$

Step 8.2

Rewrite $- \frac{2}{3} (\arcsin (\frac{u}{4}) + C)$ as $- \frac{2}{3} \arcsin (\frac{1}{4} u) + C$ .

$- \frac{2}{3} \arcsin (\frac{1}{4} u) + C$

Step 9

Replace all occurrences of $u$ with $6 x$ .

$- \frac{2}{3} \arcsin (\frac{1}{4} (6 x)) + C$

Step 10

Combine $\arcsin (\frac{1}{4} (6 x))$ and $\frac{2}{3}$ .

$- \frac{\arcsin (\frac{1}{4} \cdot 6 x) \cdot 2}{3} + C$

Step 11

Reorder terms.

$- \frac{2}{3} \arcsin (\frac{1}{4} \cdot 6 x) + C$