Calculus Examples

$\int (- \frac{4}{\sqrt{3 - 3 x^{2}}}) d x$

Step 1

Remove parentheses.

$\int - \frac{4}{\sqrt{3 - 3 x^{2}}} d x$

Step 2

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

$- \int \frac{4}{\sqrt{3 - 3 x^{2}}} d x$

Step 3

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

$- (4 \int \frac{1}{\sqrt{3 - 3 x^{2}}} d x)$

Step 4

Multiply $4$ by $- 1$ .

$- 4 \int \frac{1}{\sqrt{3 - 3 x^{2}}} d x$

Step 5

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$- 4 \int \frac{1}{\sqrt{{\sqrt{3}}^{2} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6

Simplify terms.

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Step 6.1

Simplify $\sqrt{{\sqrt{3}}^{2} - 3 \sin^{2} (t)}$ .

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Step 6.1.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 4 \int \frac{1}{\sqrt{{(3^{\frac{1}{2}})}^{2} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 \int \frac{1}{\sqrt{3^{\frac{1}{2} \cdot 2} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$- 4 \int \frac{1}{\sqrt{3^{\frac{2}{2}} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.1.4

Cancel the common factor of $2$ .

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Step 6.1.1.4.1

Cancel the common factor.

$- 4 \int \frac{1}{\sqrt{3^{\frac{2}{2}} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.1.4.2

Rewrite the expression.

$- 4 \int \frac{1}{\sqrt{3^{1} - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.1.5

Evaluate the exponent.

$- 4 \int \frac{1}{\sqrt{3 - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.2

Factor $3$ out of $3$ .

$- 4 \int \frac{1}{\sqrt{3 (1) - 3 \sin^{2} (t)}} \cos (t) d t$

Step 6.1.3

Factor $3$ out of $- 3 \sin^{2} (t)$ .

$- 4 \int \frac{1}{\sqrt{3 (1) + 3 (- \sin^{2} (t))}} \cos (t) d t$

Step 6.1.4

Factor $3$ out of $3 (1) + 3 (- \sin^{2} (t))$ .

$- 4 \int \frac{1}{\sqrt{3 (1 - \sin^{2} (t))}} \cos (t) d t$

Step 6.1.5

Apply pythagorean identity.

$- 4 \int \frac{1}{\sqrt{3 \cos^{2} (t)}} \cos (t) d t$

Step 6.1.6

Reorder $3$ and $\cos^{2} (t)$ .

$- 4 \int \frac{1}{\sqrt{\cos^{2} (t) \cdot 3}} \cos (t) d t$

Step 6.1.7

Pull terms out from under the radical.

$- 4 \int \frac{1}{\cos (t) \sqrt{3}} \cos (t) d t$

Step 6.2

Cancel the common factor of $\cos (t)$ .

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Step 6.2.1

Factor $\cos (t)$ out of $\cos (t) \sqrt{3}$ .

$- 4 \int \frac{1}{\cos (t) (\sqrt{3})} \cos (t) d t$

Step 6.2.2

Cancel the common factor.

$- 4 \int \frac{1}{\cos (t) \sqrt{3}} \cos (t) d t$

Step 6.2.3

Rewrite the expression.

$- 4 \int \frac{1}{\sqrt{3}} d t$

Step 7

Apply the constant rule.

$- 4 (\frac{1}{\sqrt{3}} t + C)$

Step 8

Simplify the answer.

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Step 8.1

Rewrite $- 4 (\frac{1}{\sqrt{3}} t + C)$ as $- 4 (\frac{\sqrt{3}}{3} t) + C$ .

$- 4 (\frac{\sqrt{3}}{3} t) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{\sqrt{3}}{3}$ and $t$ .

$- 4 \frac{\sqrt{3} t}{3} + C$

Step 8.2.2

Combine $- 4$ and $\frac{\sqrt{3} t}{3}$ .

$\frac{- 4 (\sqrt{3} t)}{3} + C$

Step 8.2.3

Move the negative in front of the fraction.

$- \frac{4 \sqrt{3} t}{3} + C$

Step 8.3

Replace all occurrences of $t$ with $\arcsin (x)$ .

$- \frac{4 \sqrt{3} \arcsin (x)}{3} + C$

Step 8.4

Reorder terms.

$- \frac{4 \sqrt{3}}{3} \arcsin (x) + C$