Calculus Examples

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

Step 1

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

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

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 $u = \csc (x)$ .

$u_{lower} = \csc (- \frac{π}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$u_{lower} = \csc (\frac{3 π}{2})$

Step 2.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$u_{lower} = - \csc (\frac{π}{2})$

Step 2.3.3

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

$u_{lower} = - 1 \cdot 1$

Step 2.3.4

Multiply $- 1$ by $1$ .

$u_{lower} = - 1$

Step 2.4

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

$u_{upper} = \csc (- \frac{π}{4})$

Step 2.5

Simplify.

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Step 2.5.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$u_{upper} = \csc (\frac{7 π}{4})$

Step 2.5.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$u_{upper} = - \csc (\frac{π}{4})$

Step 2.5.3

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

$u_{upper} = - \sqrt{2}$

Step 2.6

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

$u_{lower} = - 1$

$u_{upper} = - \sqrt{2}$

Step 2.7

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $- 1$ by $- 2$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Substitute and simplify.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

Factor $- 1$ out of $- \sqrt{2}$ .

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

Step 8.2.2

Apply the product rule to $- (\sqrt{2})$ .

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

Step 8.2.3

Raise $- 1$ to the power of $2$ .

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

Step 8.2.4

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

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Step 8.2.4.1

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

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

Step 8.2.4.2

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

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

Step 8.2.4.3

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

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

Step 8.2.4.4

Cancel the common factor of $2$ .

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Step 8.2.4.4.1

Cancel the common factor.

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

Step 8.2.4.4.2

Rewrite the expression.

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

Step 8.2.4.5

Evaluate the exponent.

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

Step 8.2.5

Multiply $2$ by $1$ .

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

Step 8.2.6

Cancel the common factor of $2$ .

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Step 8.2.6.1

Cancel the common factor.

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

Step 8.2.6.2

Rewrite the expression.

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

Step 8.2.7

Raise $- 1$ to the power of $2$ .

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

Step 8.2.8

Write $1$ as a fraction with a common denominator.

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

Step 8.2.9

Combine the numerators over the common denominator.

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

Step 8.2.10

Subtract $1$ from $2$ .

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

Step 8.2.11

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

$\frac{2}{2}$

Step 8.2.12

Cancel the common factor of $2$ .

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Step 8.2.12.1

Cancel the common factor.

$\frac{2}{2}$

Step 8.2.12.2

Rewrite the expression.

$1$