Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} - 3 \cot (x) \csc^{2} (x) d x$

Step 1

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- 3 \int_{\frac{π}{4}}^{\frac{π}{2}} \cot (x) \csc^{2} (x) d x$

Step 2

Let $u = \csc (x)$ . Then $d u = - \cot (x) \csc (x) d x$ , so $\frac{1}{- \cot (x) \csc (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \csc (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\csc (x)$ .

$\frac{d}{d x} [\csc (x)]$

Step 2.1.2

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$- \csc (x) \cot (x)$

Step 2.1.3

Reorder the factors of $- \csc (x) \cot (x)$ .

$- \cot (x) \csc (x)$

Step 2.2

Substitute the lower limit in for $x$ in $u = \csc (x)$ .

$u_{lower} = \csc (\frac{π}{4})$

Step 2.3

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$u_{lower} = \sqrt{2}$

Step 2.4

Substitute the upper limit in for $x$ in $u = \csc (x)$ .

$u_{upper} = \csc (\frac{π}{2})$

Step 2.5

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = \sqrt{2}$

$u_{upper} = 1$

Step 2.7

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

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

Step 3

Rewrite $- 1 u$ as $- u$ .

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

Step 4

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

$- 3 (- \int_{\sqrt{2}}^{1} u d u)$

Step 5

Multiply $- 1$ by $- 3$ .

$3 \int_{\sqrt{2}}^{1} u d u$

Step 6

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$3 (\frac{1}{2} u^{2}]_{\sqrt{2}}^{1})$

Step 7

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

$3 (\frac{u^{2}}{2}]_{\sqrt{2}}^{1})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{u^{2}}{2}$ at $1$ and at $\sqrt{2}$ .

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

Step 8.2

Simplify.

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Step 8.2.1

One to any power is one.

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

Step 8.2.2

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

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Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.2.3

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

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

Step 8.2.2.4

Cancel the common factor of $2$ .

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Step 8.2.2.4.1

Cancel the common factor.

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

Step 8.2.2.4.2

Rewrite the expression.

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

Step 8.2.2.5

Evaluate the exponent.

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

Step 8.2.3

Cancel the common factor of $2$ .

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Step 8.2.3.1

Cancel the common factor.

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

Step 8.2.3.2

Rewrite the expression.

$3 (\frac{1}{2} - 1 \cdot 1)$

Step 8.2.4

Multiply $- 1$ by $1$ .

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

Step 8.2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 (\frac{1}{2} - 1 \cdot \frac{2}{2})$

Step 8.2.6

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

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

Step 8.2.7

Combine the numerators over the common denominator.

$3 \frac{1 - 1 \cdot 2}{2}$

Step 8.2.8

Simplify the numerator.

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Step 8.2.8.1

Multiply $- 1$ by $2$ .

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

Step 8.2.8.2

Subtract $2$ from $1$ .

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

Step 8.2.9

Move the negative in front of the fraction.

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

Step 8.2.10

Multiply $- 1$ by $3$ .

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

Step 8.2.11

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

$\frac{- 3}{2}$

Step 8.2.12

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$

Mixed Number Form:

$- 1 \frac{1}{2}$