Calculus Examples

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

Step 1

Rewrite $\sqrt{\frac{2}{x}}$ as $\frac{\sqrt{2}}{\sqrt{x}}$ .

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

Step 2

Since $\sqrt{2}$ is constant with respect to $x$ , move $\sqrt{2}$ out of the integral.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

$\sqrt{2} \int_{1}^{4} x^{\frac{1}{2} \cdot - 1} d x$

Step 3.3.2

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

$\sqrt{2} \int_{1}^{4} x^{\frac{- 1}{2}} d x$

Step 3.3.3

Move the negative in front of the fraction.

$\sqrt{2} \int_{1}^{4} x^{- \frac{1}{2}} d x$

Step 4

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

$\sqrt{2} 2 x^{\frac{1}{2}}]_{1}^{4}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $2 x^{\frac{1}{2}}$ at $4$ and at $1$ .

$\sqrt{2} ((2 \cdot 4^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Rewrite the expression.

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

Step 5.2.4

Evaluate the exponent.

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

Step 5.2.5

Multiply $2$ by $2$ .

$\sqrt{2} (4 - 2 \cdot 1^{\frac{1}{2}})$

Step 5.2.6

One to any power is one.

$\sqrt{2} (4 - 2 \cdot 1)$

Step 5.2.7

Multiply $- 2$ by $1$ .

$\sqrt{2} (4 - 2)$

Step 5.2.8

Subtract $2$ from $4$ .

$\sqrt{2} \cdot 2$

Step 5.2.9

Move $2$ to the left of $\sqrt{2}$ .

$2 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$

Step 7