Calculus Examples

$\int_{1}^{4} \frac{\cos (\sqrt{x})}{\sqrt{x}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\int_{1}^{4} \frac{\cos (x^{\frac{1}{2}})}{\sqrt{x}} d x$

Step 1.2

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

$\int_{1}^{4} \frac{\cos (x^{\frac{1}{2}})}{x^{\frac{1}{2}}} d x$

Step 1.3

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

$\int_{1}^{4} \cos (x^{\frac{1}{2}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 1.4

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

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Step 1.4.1

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

$\int_{1}^{4} \cos (x^{\frac{1}{2}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 1.4.2

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

$\int_{1}^{4} \cos (x^{\frac{1}{2}}) x^{\frac{- 1}{2}} d x$

Step 1.4.3

Move the negative in front of the fraction.

$\int_{1}^{4} \cos (x^{\frac{1}{2}}) x^{- \frac{1}{2}} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 2.1.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.1.4

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

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

Step 2.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $- 1$ by $2$ .

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

Step 2.1.6.2

Subtract $2$ from $1$ .

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

Step 2.1.7

Move the negative in front of the fraction.

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

Step 2.1.8

Simplify.

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Step 2.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.8.2

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

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 2.2

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

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

Step 2.3

One to any power is one.

$u_{lower} = 1$

Step 2.4

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

$u_{upper} = 4^{\frac{1}{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite $4$ as $2^{2}$ .

$u_{upper} = {(2^{2})}^{\frac{1}{2}}$

Step 2.5.2

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

$u_{upper} = 2^{2 (\frac{1}{2})}$

Step 2.5.3

Cancel the common factor of $2$ .

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Step 2.5.3.1

Cancel the common factor.

$u_{upper} = 2^{2 (\frac{1}{2})}$

Step 2.5.3.2

Rewrite the expression.

$u_{upper} = 2^{1}$

Step 2.5.4

Evaluate the exponent.

$u_{upper} = 2$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 2.7

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

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

Step 3

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

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

Step 4

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$2 (\sin (u)]_{1}^{2})$

Step 5

Evaluate $\sin (u)$ at $2$ and at $1$ .

$2 (\sin (2) - \sin (1))$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Evaluate $\sin (2)$ .

$2 (0.90929742 - \sin (1))$

Step 6.1.2

Evaluate $\sin (1)$ .

$2 (0.90929742 - 1 \cdot 0.84147098)$

Step 6.1.3

Multiply $- 1$ by $0.84147098$ .

$2 (0.90929742 - 0.84147098)$

Step 6.2

Subtract $0.84147098$ from $0.90929742$ .

$2 \cdot 0.06782644$

Step 6.3

Multiply $2$ by $0.06782644$ .

$0.13565288$