Calculus Examples

$\int_{1}^{25} \frac{x - 2}{\sqrt{x}} d x$

Step 1

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

$\int_{1}^{25} \frac{x - 2}{x^{\frac{1}{2}}} d x$

Step 2

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

$\int_{1}^{25} (x - 2) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 3

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

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Step 3.1

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

$\int_{1}^{25} (x - 2) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int_{1}^{25} (x - 2) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{25} (x - 2) x^{- \frac{1}{2}} d x$

Step 4

Expand $(x - 2) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

$\int_{1}^{25} x \cdot x^{- \frac{1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\int_{1}^{25} x^{1} x^{- \frac{1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{1}^{25} x^{1 - \frac{1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 4.4

Write $1$ as a fraction with a common denominator.

$\int_{1}^{25} x^{\frac{2}{2} - \frac{1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 4.5

Combine the numerators over the common denominator.

$\int_{1}^{25} x^{\frac{2 - 1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 4.6

Subtract $1$ from $2$ .

$\int_{1}^{25} x^{\frac{1}{2}} - 2 x^{- \frac{1}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int_{1}^{25} x^{\frac{1}{2}} d x + \int_{1}^{25} - 2 x^{- \frac{1}{2}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{25} + \int_{1}^{25} - 2 x^{- \frac{1}{2}} d x$

Step 7

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

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{25} - 2 \int_{1}^{25} x^{- \frac{1}{2}} d x$

Step 8

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

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{25} - 2 (2 x^{\frac{1}{2}}]_{1}^{25})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $25$ and at $1$ .

$(\frac{2}{3} \cdot 25^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 x^{\frac{1}{2}}]_{1}^{25})$

Step 9.2

Evaluate $2 x^{\frac{1}{2}}$ at $25$ and at $1$ .

$\frac{2}{3} \cdot 25^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3

Simplify.

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Step 9.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{2}{3} \cdot {(5^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.2

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

$\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.3

Cancel the common factor of $2$ .

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Step 9.3.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 5^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.4

Raise $5$ to the power of $3$ .

$\frac{2}{3} \cdot 125 - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.5

Combine $\frac{2}{3}$ and $125$ .

$\frac{2 \cdot 125}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6

Multiply $2$ by $125$ .

$\frac{250}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.7

One to any power is one.

$\frac{250}{3} - \frac{2}{3} \cdot 1 - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.8

Multiply $- 1$ by $1$ .

$\frac{250}{3} - \frac{2}{3} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.9

Combine the numerators over the common denominator.

$\frac{250 - 2}{3} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.10

Subtract $2$ from $250$ .

$\frac{248}{3} - 2 (2 \cdot 25^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11

Rewrite $25$ as $5^{2}$ .

$\frac{248}{3} - 2 (2 \cdot {(5^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.12

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

$\frac{248}{3} - 2 (2 \cdot 5^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.13

Cancel the common factor of $2$ .

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Step 9.3.13.1

Cancel the common factor.

$\frac{248}{3} - 2 (2 \cdot 5^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.13.2

Rewrite the expression.

$\frac{248}{3} - 2 (2 \cdot 5^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.14

Evaluate the exponent.

$\frac{248}{3} - 2 (2 \cdot 5 - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.15

Multiply $2$ by $5$ .

$\frac{248}{3} - 2 (10 - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.16

One to any power is one.

$\frac{248}{3} - 2 (10 - 2 \cdot 1)$

Step 9.3.17

Multiply $- 2$ by $1$ .

$\frac{248}{3} - 2 (10 - 2)$

Step 9.3.18

Subtract $2$ from $10$ .

$\frac{248}{3} - 2 \cdot 8$

Step 9.3.19

Multiply $- 2$ by $8$ .

$\frac{248}{3} - 16$

Step 9.3.20

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{248}{3} - 16 \cdot \frac{3}{3}$

Step 9.3.21

Combine $- 16$ and $\frac{3}{3}$ .

$\frac{248}{3} + \frac{- 16 \cdot 3}{3}$

Step 9.3.22

Combine the numerators over the common denominator.

$\frac{248 - 16 \cdot 3}{3}$

Step 9.3.23

Simplify the numerator.

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Step 9.3.23.1

Multiply $- 16$ by $3$ .

$\frac{248 - 48}{3}$

Step 9.3.23.2

Subtract $48$ from $248$ .

$\frac{200}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{200}{3}$

Decimal Form:

$66.\overline{6}$

Mixed Number Form:

$66 \frac{2}{3}$

Step 11