Calculus Examples

$\int_{1}^{16} \frac{x - 5}{\sqrt{x}} d x$

Step 1

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

$\int_{1}^{16} \frac{x - 5}{x^{\frac{1}{2}}} d x$

Step 2

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

$\int_{1}^{16} (x - 5) {(x^{\frac{1}{2}})}^{- 1} d x$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{1}^{16} (x - 5) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int_{1}^{16} (x - 5) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{16} (x - 5) x^{- \frac{1}{2}} d x$

Step 4

Expand $(x - 5) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

$\int_{1}^{16} x \cdot x^{- \frac{1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 4.2

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

$\int_{1}^{16} x^{1} x^{- \frac{1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 4.3

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

$\int_{1}^{16} x^{1 - \frac{1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 4.4

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

$\int_{1}^{16} x^{\frac{2}{2} - \frac{1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 4.5

Combine the numerators over the common denominator.

$\int_{1}^{16} x^{\frac{2 - 1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 4.6

Subtract $1$ from $2$ .

$\int_{1}^{16} x^{\frac{1}{2}} - 5 x^{- \frac{1}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int_{1}^{16} x^{\frac{1}{2}} d x + \int_{1}^{16} - 5 x^{- \frac{1}{2}} d x$

Step 6

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

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{16} + \int_{1}^{16} - 5 x^{- \frac{1}{2}} d x$

Step 7

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{16} - 5 \int_{1}^{16} x^{- \frac{1}{2}} d x$

Step 8

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

$\frac{2}{3} x^{\frac{3}{2}}]_{1}^{16} - 5 (2 x^{\frac{1}{2}}]_{1}^{16})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $16$ and at $1$ .

$(\frac{2}{3} \cdot 16^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 x^{\frac{1}{2}}]_{1}^{16})$

Step 9.2

Evaluate $2 x^{\frac{1}{2}}$ at $16$ and at $1$ .

$\frac{2}{3} \cdot 16^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3

Simplify.

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Step 9.3.1

Rewrite $16$ as $4^{2}$ .

$\frac{2}{3} \cdot {(4^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.2

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

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.3

Cancel the common factor of $2$ .

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Step 9.3.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 4^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.4

Raise $4$ to the power of $3$ .

$\frac{2}{3} \cdot 64 - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.5

Combine $\frac{2}{3}$ and $64$ .

$\frac{2 \cdot 64}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6

Multiply $2$ by $64$ .

$\frac{128}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.7

One to any power is one.

$\frac{128}{3} - \frac{2}{3} \cdot 1 - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.8

Multiply $- 1$ by $1$ .

$\frac{128}{3} - \frac{2}{3} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.9

Combine the numerators over the common denominator.

$\frac{128 - 2}{3} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.10

Subtract $2$ from $128$ .

$\frac{126}{3} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11

Cancel the common factor of $126$ and $3$ .

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Step 9.3.11.1

Factor $3$ out of $126$ .

$\frac{3 \cdot 42}{3} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11.2

Cancel the common factors.

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Step 9.3.11.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 42}{3 (1)} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11.2.2

Cancel the common factor.

$\frac{3 \cdot 42}{3 \cdot 1} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11.2.3

Rewrite the expression.

$\frac{42}{1} - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.11.2.4

Divide $42$ by $1$ .

$42 - 5 (2 \cdot 16^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.12

Rewrite $16$ as $4^{2}$ .

$42 - 5 (2 \cdot {(4^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.13

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

$42 - 5 (2 \cdot 4^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.14

Cancel the common factor of $2$ .

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Step 9.3.14.1

Cancel the common factor.

$42 - 5 (2 \cdot 4^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.14.2

Rewrite the expression.

$42 - 5 (2 \cdot 4^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.15

Evaluate the exponent.

$42 - 5 (2 \cdot 4 - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.16

Multiply $2$ by $4$ .

$42 - 5 (8 - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.17

One to any power is one.

$42 - 5 (8 - 2 \cdot 1)$

Step 9.3.18

Multiply $- 2$ by $1$ .

$42 - 5 (8 - 2)$

Step 9.3.19

Subtract $2$ from $8$ .

$42 - 5 \cdot 6$

Step 9.3.20

Multiply $- 5$ by $6$ .

$42 - 30$

Step 9.3.21

Subtract $30$ from $42$ .

$12$

Step 10