Calculus Examples

$\int_{0}^{9} \frac{1}{\sqrt{x}} d x$

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

$\int_{0}^{9} \frac{1}{x^{\frac{1}{2}}} d x$

Step 1.2

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

$\int_{0}^{9} {(x^{\frac{1}{2}})}^{- 1} d x$

Step 1.3

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

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Step 1.3.1

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

$\int_{0}^{9} x^{\frac{1}{2} \cdot - 1} d x$

Step 1.3.2

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

$\int_{0}^{9} x^{\frac{- 1}{2}} d x$

Step 1.3.3

Move the negative in front of the fraction.

$\int_{0}^{9} x^{- \frac{1}{2}} d x$

Step 2

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

$2 x^{\frac{1}{2}}]_{0}^{9}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $2 x^{\frac{1}{2}}$ at $9$ and at $0$ .

$(2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $9$ as $3^{2}$ .

$2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.2

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

$2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

$2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.3.2

Rewrite the expression.

$2 \cdot 3^{1} - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.4

Evaluate the exponent.

$2 \cdot 3 - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.5

Multiply $2$ by $3$ .

$6 - 2 \cdot 0^{\frac{1}{2}}$

Step 3.2.6

Rewrite $0$ as $0^{2}$ .

$6 - 2 \cdot {(0^{2})}^{\frac{1}{2}}$

Step 3.2.7

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

$6 - 2 \cdot 0^{2 (\frac{1}{2})}$

Step 3.2.8

Cancel the common factor of $2$ .

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Step 3.2.8.1

Cancel the common factor.

$6 - 2 \cdot 0^{2 (\frac{1}{2})}$

Step 3.2.8.2

Rewrite the expression.

$6 - 2 \cdot 0^{1}$

Step 3.2.9

Evaluate the exponent.

$6 - 2 \cdot 0$

Step 3.2.10

Multiply $- 2$ by $0$ .

$6 + 0$

Step 3.2.11

Add $6$ and $0$ .

$6$

Step 4