Calculus Examples

$\int_{\frac{π}{3}}^{\frac{π}{2}} \frac{\sin (x)}{\sqrt{1 + \cos (x)}} d x$

Step 1

Let $u = 1 + \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- d u = \sin (x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 + \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + \cos (x)$ .

$\frac{d}{d x} [1 + \cos (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\cos (x)]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [\cos (x)]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\cos (x)]$

Step 1.1.3

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

$0 - \sin (x)$

Step 1.1.4

Subtract $\sin (x)$ from $0$ .

$- \sin (x)$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + \cos (x)$ .

$u_{lower} = 1 + \cos (\frac{π}{3})$

Step 1.3

Simplify.

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Step 1.3.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{lower} = 1 + \frac{1}{2}$

Step 1.3.2

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

$u_{lower} = \frac{2}{2} + \frac{1}{2}$

Step 1.3.3

Combine the numerators over the common denominator.

$u_{lower} = \frac{2 + 1}{2}$

Step 1.3.4

Add $2$ and $1$ .

$u_{lower} = \frac{3}{2}$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + \cos (x)$ .

$u_{upper} = 1 + \cos (\frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{upper} = 1 + 0$

Step 1.5.2

Add $1$ and $0$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = \frac{3}{2}$

$u_{upper} = 1$

Step 1.7

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $2 u^{\frac{1}{2}}$ at $1$ and at $\frac{3}{2}$ .

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

Step 6.2

Simplify.

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Step 6.2.1

One to any power is one.

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

Step 6.2.2

Multiply $2$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $- 1$ by $2$ .

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

Step 7.3

Multiply $- 2$ by $- 1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.44948974 \dots$