Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} \frac{\cos (x)}{\sin^{2} (x)} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\sin (x)$ .

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

Step 1.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 1.2

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

$u_{lower} = \sin (\frac{π}{4})$

Step 1.3

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

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

Step 1.4

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

$u_{upper} = \sin (\frac{π}{2})$

Step 1.5

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

$u_{upper} = 1$

Step 1.6

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

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

$u_{upper} = 1$

Step 1.7

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

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

Step 4.2.2

Multiply $- 1$ by $1$ .

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

Step 4.2.3

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

Combine and simplify the denominator.

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Step 5.2.1

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

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

Step 5.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.4

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

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

Step 5.2.5

Add $1$ and $1$ .

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

Step 5.2.6

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

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Step 5.2.6.1

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

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

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

Step 5.2.6.4

Cancel the common factor of $2$ .

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Step 5.2.6.4.1

Cancel the common factor.

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

Step 5.2.6.4.2

Rewrite the expression.

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

Step 5.2.6.5

Evaluate the exponent.

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

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Divide $\sqrt{2}$ by $1$ .

$- 1 + \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 1 + \sqrt{2}$

Decimal Form:

$0.41421356 \dots$