Calculus Examples

$\int_{0}^{81} \frac{81}{\sqrt{x}} d x$

Step 1

Since $81$ is constant with respect to $x$ , move $81$ out of the integral.

$81 \int_{0}^{81} \frac{1}{\sqrt{x}} d x$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

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

$81 \int_{0}^{81} \frac{1}{x^{\frac{1}{2}}} d x$

Step 2.2

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

$81 \int_{0}^{81} {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$81 \int_{0}^{81} x^{\frac{1}{2} \cdot - 1} d x$

Step 2.3.2

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

$81 \int_{0}^{81} x^{\frac{- 1}{2}} d x$

Step 2.3.3

Move the negative in front of the fraction.

$81 \int_{0}^{81} x^{- \frac{1}{2}} d x$

Step 3

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

$81 (2 x^{\frac{1}{2}}]_{0}^{81})$

Step 4

Substitute and simplify.

Tap for more steps...

Step 4.1

Evaluate $2 x^{\frac{1}{2}}$ at $81$ and at $0$ .

$81 ((2 \cdot 81^{\frac{1}{2}}) - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Rewrite $81$ as $9^{2}$ .

$81 (2 \cdot {(9^{2})}^{\frac{1}{2}} - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.2

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

$81 (2 \cdot 9^{2 (\frac{1}{2})} - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.1

Cancel the common factor.

$81 (2 \cdot 9^{2 (\frac{1}{2})} - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.3.2

Rewrite the expression.

$81 (2 \cdot 9^{1} - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.4

Evaluate the exponent.

$81 (2 \cdot 9 - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.5

Multiply $2$ by $9$ .

$81 (18 - 2 \cdot 0^{\frac{1}{2}})$

Step 4.2.6

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

$81 (18 - 2 \cdot {(0^{2})}^{\frac{1}{2}})$

Step 4.2.7

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

$81 (18 - 2 \cdot 0^{2 (\frac{1}{2})})$

Step 4.2.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.8.1

Cancel the common factor.

$81 (18 - 2 \cdot 0^{2 (\frac{1}{2})})$

Step 4.2.8.2

Rewrite the expression.

$81 (18 - 2 \cdot 0^{1})$

Step 4.2.9

Evaluate the exponent.

$81 (18 - 2 \cdot 0)$

Step 4.2.10

Multiply $- 2$ by $0$ .

$81 (18 + 0)$

Step 4.2.11

Add $18$ and $0$ .

$81 \cdot 18$

Step 4.2.12

Multiply $81$ by $18$ .

$1458$

Step 5