Calculus Examples

$\int_{\frac{π}{3}}^{\frac{2 π}{3}} \csc^{2} (\frac{1}{2} t) d t$

Step 1

Let $u = \frac{1}{2} t$ . Then $d u = \frac{1}{2} d t$ , so $2 d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{1}{2} t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $\frac{1}{2} t$ .

$\frac{d}{d t} [\frac{1}{2} t]$

Step 1.1.2

Since $\frac{1}{2}$ is constant with respect to $t$ , the derivative of $\frac{1}{2} t$ with respect to $t$ is $\frac{1}{2} \frac{d}{d t} [t]$ .

$\frac{1}{2} \frac{d}{d t} [t]$

Step 1.1.3

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

$\frac{1}{2} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 1.2

Substitute the lower limit in for $t$ in $u = \frac{1}{2} t$ .

$u_{lower} = \frac{1}{2} \cdot \frac{π}{3}$

Step 1.3

Multiply $\frac{1}{2} \cdot \frac{π}{3}$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{π}{3}$ .

$u_{lower} = \frac{π}{2 \cdot 3}$

Step 1.3.2

Multiply $2$ by $3$ .

$u_{lower} = \frac{π}{6}$

Step 1.4

Substitute the upper limit in for $t$ in $u = \frac{1}{2} t$ .

$u_{upper} = \frac{1}{2} \cdot \frac{2 π}{3}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $2 π$ .

$u_{upper} = \frac{1}{2} \cdot \frac{2 (π)}{3}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = \frac{1}{2} \cdot \frac{2 π}{3}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{π}{6}$

$u_{upper} = \frac{π}{3}$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{\frac{π}{6}}^{\frac{π}{3}} \csc^{2} (u) \frac{1}{\frac{1}{2}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int_{\frac{π}{6}}^{\frac{π}{3}} \csc^{2} (u) (1 \cdot 2) d u$

Step 2.2

Multiply $2$ by $1$ .

$\int_{\frac{π}{6}}^{\frac{π}{3}} \csc^{2} (u) \cdot 2 d u$

Step 2.3

Move $2$ to the left of $\csc^{2} (u)$ .

$\int_{\frac{π}{6}}^{\frac{π}{3}} 2 \csc^{2} (u) d u$

Step 3

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int_{\frac{π}{6}}^{\frac{π}{3}} \csc^{2} (u) d u$

Step 4

Since the derivative of $- \cot (u)$ is $\csc^{2} (u)$ , the integral of $\csc^{2} (u)$ is $- \cot (u)$ .

$2 (- \cot (u)]_{\frac{π}{6}}^{\frac{π}{3}})$

Step 5

Evaluate $- \cot (u)$ at $\frac{π}{3}$ and at $\frac{π}{6}$ .

$2 (- \cot (\frac{π}{3}) + \cot (\frac{π}{6}))$

Step 6

Simplify.

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Step 6.1

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$2 (- \frac{1}{\sqrt{3}} + \cot (\frac{π}{6}))$

Step 6.2

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$2 (- \frac{1}{\sqrt{3}} + \sqrt{3})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (- \frac{1}{\sqrt{3}}) + 2 \sqrt{3}$

Step 7.2

Multiply $2 (- \frac{1}{\sqrt{3}})$ .

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Step 7.2.1

Multiply $- 1$ by $2$ .

$- 2 \frac{1}{\sqrt{3}} + 2 \sqrt{3}$

Step 7.2.2

Combine $- 2$ and $\frac{1}{\sqrt{3}}$ .

$\frac{- 2}{\sqrt{3}} + 2 \sqrt{3}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{2}{\sqrt{3}} + 2 \sqrt{3}$

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + 2 \sqrt{3}$

Step 8.1.2

Combine and simplify the denominator.

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Step 8.1.2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} + 2 \sqrt{3}$

Step 8.1.2.2

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

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + 2 \sqrt{3}$

Step 8.1.2.3

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

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + 2 \sqrt{3}$

Step 8.1.2.4

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

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + 2 \sqrt{3}$

Step 8.1.2.5

Add $1$ and $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} + 2 \sqrt{3}$

Step 8.1.2.6

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

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Step 8.1.2.6.1

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

$- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + 2 \sqrt{3}$

Step 8.1.2.6.2

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

$- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + 2 \sqrt{3}$

Step 8.1.2.6.3

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

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 2 \sqrt{3}$

Step 8.1.2.6.4

Cancel the common factor of $2$ .

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Step 8.1.2.6.4.1

Cancel the common factor.

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 2 \sqrt{3}$

Step 8.1.2.6.4.2

Rewrite the expression.

$- \frac{2 \sqrt{3}}{3^{1}} + 2 \sqrt{3}$

Step 8.1.2.6.5

Evaluate the exponent.

$- \frac{2 \sqrt{3}}{3} + 2 \sqrt{3}$

Step 8.2

To write $2 \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 \sqrt{3}}{3} + 2 \sqrt{3} \cdot \frac{3}{3}$

Step 8.3

Combine $2 \sqrt{3}$ and $\frac{3}{3}$ .

$- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3} \cdot 3}{3}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{- 2 \sqrt{3} + 2 \sqrt{3} \cdot 3}{3}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $3$ by $2$ .

$\frac{- 2 \sqrt{3} + 6 \sqrt{3}}{3}$

Step 8.5.2

Add $- 2 \sqrt{3}$ and $6 \sqrt{3}$ .

$\frac{4 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{3}}{3}$

Decimal Form:

$2.30940107 \dots$