Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$\frac{1}{2} x^{2}]_{1}^{9} + \int_{1}^{9} - \frac{1}{\sqrt{x}} d x$

Step 3

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

$\frac{1}{2} x^{2}]_{1}^{9} - \int_{1}^{9} \frac{1}{\sqrt{x}} d x$

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

$\frac{1}{2} x^{2}]_{1}^{9} - (2 x^{\frac{1}{2}}]_{1}^{9})$

Step 6

Substitute and simplify.

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Step 6.1

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

$(\frac{1}{2} \cdot 9^{2}) - \frac{1}{2} \cdot 1^{2} - (2 x^{\frac{1}{2}}]_{1}^{9})$

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $9$ to the power of $2$ .

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

Step 6.3.2

Combine $\frac{1}{2}$ and $81$ .

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

Step 6.3.3

One to any power is one.

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

Step 6.3.4

Multiply $- 1$ by $1$ .

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

Step 6.3.5

Combine the numerators over the common denominator.

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

Step 6.3.6

Subtract $1$ from $81$ .

$\frac{80}{2} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.7

Cancel the common factor of $80$ and $2$ .

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Step 6.3.7.1

Factor $2$ out of $80$ .

$\frac{2 \cdot 40}{2} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.7.2

Cancel the common factors.

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Step 6.3.7.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 40}{2 (1)} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.7.2.2

Cancel the common factor.

$\frac{2 \cdot 40}{2 \cdot 1} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.7.2.3

Rewrite the expression.

$\frac{40}{1} - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.7.2.4

Divide $40$ by $1$ .

$40 - (2 \cdot 9^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.8

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

$40 - (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.9

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

$40 - (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.10

Cancel the common factor of $2$ .

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Step 6.3.10.1

Cancel the common factor.

$40 - (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.10.2

Rewrite the expression.

$40 - (2 \cdot 3^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.11

Evaluate the exponent.

$40 - (2 \cdot 3 - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.12

Multiply $2$ by $3$ .

$40 - (6 - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.13

One to any power is one.

$40 - (6 - 2 \cdot 1)$

Step 6.3.14

Multiply $- 2$ by $1$ .

$40 - (6 - 2)$

Step 6.3.15

Subtract $2$ from $6$ .

$40 - 1 \cdot 4$

Step 6.3.16

Multiply $- 1$ by $4$ .

$40 - 4$

Step 6.3.17

Subtract $4$ from $40$ .

$36$

Step 7