Calculus Examples

$\int_{1}^{\sqrt{x}} 18 t^{9} d t$

Step 1

Since $18$ is constant with respect to $t$ , move $18$ out of the integral.

$18 \int_{1}^{\sqrt{x}} t^{9} d t$

Step 2

By the Power Rule, the integral of $t^{9}$ with respect to $t$ is $\frac{1}{10} t^{10}$ .

$18 (\frac{1}{10} t^{10}]_{1}^{\sqrt{x}})$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{10}$ and $t^{10}$ .

$18 (\frac{t^{10}}{10}]_{1}^{\sqrt{x}})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{t^{10}}{10}$ at $\sqrt{x}$ and at $1$ .

$18 ((\frac{{\sqrt{x}}^{10}}{10}) - \frac{1^{10}}{10})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Rewrite ${\sqrt{x}}^{10}$ as $x^{5}$ .

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Step 3.2.2.1.1

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

$18 (\frac{{(x^{\frac{1}{2}})}^{10}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.2

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

$18 (\frac{x^{\frac{1}{2} \cdot 10}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.3

Combine $\frac{1}{2}$ and $10$ .

$18 (\frac{x^{\frac{10}{2}}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.4

Cancel the common factor of $10$ and $2$ .

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Step 3.2.2.1.4.1

Factor $2$ out of $10$ .

$18 (\frac{x^{\frac{2 \cdot 5}{2}}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.4.2

Cancel the common factors.

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Step 3.2.2.1.4.2.1

Factor $2$ out of $2$ .

$18 (\frac{x^{\frac{2 \cdot 5}{2 (1)}}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.4.2.2

Cancel the common factor.

$18 (\frac{x^{\frac{2 \cdot 5}{2 \cdot 1}}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.4.2.3

Rewrite the expression.

$18 (\frac{x^{\frac{5}{1}}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.1.4.2.4

Divide $5$ by $1$ .

$18 (\frac{x^{5}}{10} - \frac{1^{10}}{10})$

Step 3.2.2.2

One to any power is one.

$18 (\frac{x^{5}}{10} - \frac{1}{10})$

Step 3.3

Reorder terms.

$18 (\frac{1}{10} x^{5} - \frac{1}{10})$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{10}$ and $x^{5}$ .

$18 (\frac{x^{5}}{10} - \frac{1}{10})$

Step 4.2

Apply the distributive property.

$18 \frac{x^{5}}{10} + 18 (- \frac{1}{10})$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Factor $2$ out of $18$ .

$2 (9) \frac{x^{5}}{10} + 18 (- \frac{1}{10})$

Step 4.3.2

Factor $2$ out of $10$ .

$2 \cdot 9 \frac{x^{5}}{2 \cdot 5} + 18 (- \frac{1}{10})$

Step 4.3.3

Cancel the common factor.

$2 \cdot 9 \frac{x^{5}}{2 \cdot 5} + 18 (- \frac{1}{10})$

Step 4.3.4

Rewrite the expression.

$9 \frac{x^{5}}{5} + 18 (- \frac{1}{10})$

Step 4.4

Combine $9$ and $\frac{x^{5}}{5}$ .

$\frac{9 x^{5}}{5} + 18 (- \frac{1}{10})$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$\frac{9 x^{5}}{5} + 18 (\frac{- 1}{10})$

Step 4.5.2

Factor $2$ out of $18$ .

$\frac{9 x^{5}}{5} + 2 (9) \frac{- 1}{10}$

Step 4.5.3

Factor $2$ out of $10$ .

$\frac{9 x^{5}}{5} + 2 \cdot 9 \frac{- 1}{2 \cdot 5}$

Step 4.5.4

Cancel the common factor.

$\frac{9 x^{5}}{5} + 2 \cdot 9 \frac{- 1}{2 \cdot 5}$

Step 4.5.5

Rewrite the expression.

$\frac{9 x^{5}}{5} + 9 (\frac{- 1}{5})$

Step 4.6

Combine $9$ and $\frac{- 1}{5}$ .

$\frac{9 x^{5}}{5} + \frac{9 \cdot - 1}{5}$

Step 4.7

Multiply $9$ by $- 1$ .

$\frac{9 x^{5}}{5} + \frac{- 9}{5}$

Step 4.8

Move the negative in front of the fraction.

$\frac{9 x^{5}}{5} - \frac{9}{5}$