Calculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply by $1$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

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

Step 2.1.7

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

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

Step 2.1.8

Pull terms out from under the radical.

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

Step 2.2

Simplify.

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Step 2.2.1

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.2.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.4

Add $1$ and $1$ .

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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

Step 2.2.5.3

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

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

Step 2.2.5.4

Cancel the common factor of $2$ .

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Step 2.2.5.4.1

Cancel the common factor.

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

Step 2.2.5.4.2

Rewrite the expression.

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

Step 2.2.5.5

Evaluate the exponent.

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

Step 2.2.6

Combine $\sqrt{3}$ and $\frac{1}{3 \sin (t) \cos (t)}$ .

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

Step 2.2.7

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

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

Step 2.2.8

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

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Step 2.2.8.1

Cancel the common factor.

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

Step 2.2.8.2

Rewrite the expression.

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

Step 3

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

$\frac{\sqrt{3}}{3} \int \frac{1}{\sin (t)} d t$

Step 4

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

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

Step 5

The integral of $\csc (t)$ with respect to $t$ is $\ln (| \csc (t) - \cot (t) |)$ .

$\frac{\sqrt{3}}{3} (\ln (| \csc (t) - \cot (t) |) + C)$

Step 6

Simplify.

$\frac{\sqrt{3}}{3} \ln (| \csc (t) - \cot (t) |) + C$

Step 7

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

$\frac{\sqrt{3}}{3} \ln (| \csc (\arcsin (\frac{u}{\sqrt{3}})) - \cot (\arcsin (\frac{u}{\sqrt{3}})) |) + C$

Step 8

Reorder terms.

$\frac{\sqrt{3}}{3} \ln (| \csc (\arcsin (\frac{1}{\sqrt{3}} u)) - \cot (\arcsin (\frac{1}{\sqrt{3}} u)) |) + C$