Calculus Examples

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

Step 1

Let $x = \frac{\sqrt{3}}{2} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{3} \cos (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{3} \cos (t)}{2}$ is positive.

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

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{\sqrt{3}}^{2} - 4 {(\frac{\sqrt{3}}{2} \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

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

Step 2.1.1.1.2

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

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

Step 2.1.1.1.3

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

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

Step 2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.1.4.1

Cancel the common factor.

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

Step 2.1.1.1.4.2

Rewrite the expression.

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

Step 2.1.1.1.5

Evaluate the exponent.

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

Step 2.1.1.2

Combine $\frac{\sqrt{3}}{2}$ and $\sin (t)$ .

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

Step 2.1.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.3.1

Apply the product rule to $\frac{\sqrt{3} \sin (t)}{2}$ .

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

Step 2.1.1.3.2

Apply the product rule to $\sqrt{3} \sin (t)$ .

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

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

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

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

Step 2.1.1.4.3

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

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

Step 2.1.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.1.4.4.1

Cancel the common factor.

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

Step 2.1.1.4.4.2

Rewrite the expression.

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

Step 2.1.1.4.5

Evaluate the exponent.

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

Step 2.1.1.5

Raise $2$ to the power of $2$ .

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

Step 2.1.1.6

Cancel the common factor of $4$ .

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Step 2.1.1.6.1

Factor $4$ out of $- 4$ .

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

Step 2.1.1.6.2

Cancel the common factor.

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

Step 2.1.1.6.3

Rewrite the expression.

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

Step 2.1.1.7

Multiply $3$ by $- 1$ .

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

Step 2.1.2

Factor $3$ out of $3$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

Pull terms out from under the radical.

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $\frac{1}{\cos (t) \sqrt{3}}$ by $\frac{\sqrt{3} \cos (t)}{2}$ .

$\int \frac{\sqrt{3} \cos (t)}{\cos (t) \sqrt{3} \cdot 2} d t$

Step 2.2.2

Move $2$ to the left of $\cos (t) \sqrt{3}$ .

$\int \frac{\sqrt{3} \cos (t)}{2 \cdot (\cos (t) \sqrt{3})} d t$

Step 2.2.3

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.3.1

Cancel the common factor.

$\int \frac{\sqrt{3} \cos (t)}{2 \cos (t) \sqrt{3}} d t$

Step 2.2.3.2

Rewrite the expression.

$\int \frac{\cos (t)}{2 \cos (t)} d t$

Step 2.2.4

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

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Step 2.2.4.1

Cancel the common factor.

$\int \frac{\cos (t)}{2 \cos (t)} d t$

Step 2.2.4.2

Rewrite the expression.

$\int \frac{1}{2} d t$

Step 3

Apply the constant rule.

$\frac{1}{2} t + C$

Step 4

Simplify the answer.

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Step 4.1

Replace all occurrences of $t$ with $\arcsin (\frac{2 x}{\sqrt{3}})$ .

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

Step 4.2

Reorder terms.

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