Calculus Examples

$\frac{d}{(d x) \int_{0}^{x} 13 t^{2} + \sqrt{t} d t}$

Step 1

Cancel the common factor of $d$ .

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Step 1.1

Cancel the common factor.

$\frac{d}{d x \int_{0}^{x} 13 t^{2} + \sqrt{t} d t}$

Step 1.2

Rewrite the expression.

$\frac{1}{x \int_{0}^{x} 13 t^{2} + \sqrt{t} d t}$

Step 2

Split the single integral into multiple integrals.

$\frac{1}{x (\int_{0}^{x} 13 t^{2} d t + \int_{0}^{x} \sqrt{t} d t)}$

Step 3

Since $13$ is constant with respect to $t$ , move $13$ out of the integral.

$\frac{1}{x (13 \int_{0}^{x} t^{2} d t + \int_{0}^{x} \sqrt{t} d t)}$

Step 4

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

$\frac{1}{x (13 (\frac{1}{3} t^{3}]_{0}^{x}) + \int_{0}^{x} \sqrt{t} d t)}$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{3}$ and $t^{3}$ .

$\frac{1}{x (13 (\frac{t^{3}}{3}]_{0}^{x}) + \int_{0}^{x} \sqrt{t} d t)}$

Step 5.2

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

$\frac{1}{x (13 (\frac{t^{3}}{3}]_{0}^{x}) + \int_{0}^{x} t^{\frac{1}{2}} d t)}$

Step 6

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

$\frac{1}{x (13 (\frac{t^{3}}{3}]_{0}^{x}) + \frac{2}{3} t^{\frac{3}{2}}]_{0}^{x})}$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{2}{3}$ and $t^{\frac{3}{2}}$ .

$\frac{1}{x (13 (\frac{t^{3}}{3}]_{0}^{x}) + \frac{2 t^{\frac{3}{2}}}{3}]_{0}^{x})}$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{t^{3}}{3}$ at $x$ and at $0$ .

$\frac{1}{x (13 ((\frac{x^{3}}{3}) - \frac{0^{3}}{3}) + \frac{2 t^{\frac{3}{2}}}{3}]_{0}^{x})}$

Step 7.2.2

Evaluate $\frac{2 t^{\frac{3}{2}}}{3}$ at $x$ and at $0$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{0}{3}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.2

Cancel the common factor of $0$ and $3$ .

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Step 7.2.3.2.1

Factor $3$ out of $0$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{3 (0)}{3}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.2.2

Cancel the common factors.

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Step 7.2.3.2.2.1

Factor $3$ out of $3$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.2.2.2

Cancel the common factor.

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.2.2.3

Rewrite the expression.

$\frac{1}{x (13 (\frac{x^{3}}{3} - \frac{0}{1}) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.2.2.4

Divide $0$ by $1$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} - 0) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.3

Multiply $- 1$ by $0$ .

$\frac{1}{x (13 (\frac{x^{3}}{3} + 0) + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.4

Add $\frac{x^{3}}{3}$ and $0$ .

$\frac{1}{x (13 \frac{x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.5

Combine $13$ and $\frac{x^{3}}{3}$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})}$

Step 7.2.3.6

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

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3})}$

Step 7.2.3.7

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

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})}$

Step 7.2.3.8

Cancel the common factor of $2$ .

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Step 7.2.3.8.1

Cancel the common factor.

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})}$

Step 7.2.3.8.2

Rewrite the expression.

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{3}}{3})}$

Step 7.2.3.9

Raising $0$ to any positive power yields $0$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \cdot 0}{3})}$

Step 7.2.3.10

Multiply $2$ by $0$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{0}{3})}$

Step 7.2.3.11

Cancel the common factor of $0$ and $3$ .

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Step 7.2.3.11.1

Factor $3$ out of $0$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{3 (0)}{3})}$

Step 7.2.3.11.2

Cancel the common factors.

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Step 7.2.3.11.2.1

Factor $3$ out of $3$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{3 \cdot 0}{3 \cdot 1})}$

Step 7.2.3.11.2.2

Cancel the common factor.

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{3 \cdot 0}{3 \cdot 1})}$

Step 7.2.3.11.2.3

Rewrite the expression.

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - \frac{0}{1})}$

Step 7.2.3.11.2.4

Divide $0$ by $1$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} - 0)}$

Step 7.2.3.12

Multiply $- 1$ by $0$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3} + 0)}$

Step 7.2.3.13

Add $\frac{2 x^{\frac{3}{2}}}{3}$ and $0$ .

$\frac{1}{x (\frac{13 x^{3}}{3} + \frac{2 x^{\frac{3}{2}}}{3})}$

Step 7.3

Reorder terms.

$\frac{1}{x (\frac{13}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}})}$

Step 8

Simplify.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Factor $x^{\frac{3}{2}}$ out of $\frac{13}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}}$ .

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Step 8.1.1.1

Reorder the expression.

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Step 8.1.1.1.1

Reorder $\frac{13}{3}$ and $x^{3}$ .

$\frac{1}{x (x^{3} \frac{13}{3} + \frac{2}{3} x^{\frac{3}{2}})}$

Step 8.1.1.1.2

Reorder $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$\frac{1}{x (x^{3} \frac{13}{3} + x^{\frac{3}{2}} \frac{2}{3})}$

Step 8.1.1.2

Factor $x^{\frac{3}{2}}$ out of $x^{3} \frac{13}{3}$ .

$\frac{1}{x (x^{\frac{3}{2}} (x^{\frac{3}{2}} \frac{13}{3}) + x^{\frac{3}{2}} \frac{2}{3})}$

Step 8.1.1.3

Factor $x^{\frac{3}{2}}$ out of $x^{\frac{3}{2}} (x^{\frac{3}{2}} \frac{13}{3}) + x^{\frac{3}{2}} (\frac{2}{3})$ .

$\frac{1}{x (x^{\frac{3}{2}} (x^{\frac{3}{2}} \frac{13}{3} + \frac{2}{3}))}$

Step 8.1.2

Combine $x^{\frac{3}{2}}$ and $\frac{13}{3}$ .

$\frac{1}{x (x^{\frac{3}{2}} (\frac{x^{\frac{3}{2}} \cdot 13}{3} + \frac{2}{3}))}$

Step 8.1.3

Move $13$ to the left of $x^{\frac{3}{2}}$ .

$\frac{1}{x (x^{\frac{3}{2}} (\frac{13 \cdot x^{\frac{3}{2}}}{3} + \frac{2}{3}))}$

Step 8.1.4

Remove unnecessary parentheses.

$\frac{1}{x \cdot x^{\frac{3}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.5

Multiply $x$ by $x^{\frac{3}{2}}$ by adding the exponents.

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Step 8.1.5.1

Multiply $x$ by $x^{\frac{3}{2}}$ .

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Step 8.1.5.1.1

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

$\frac{1}{x^{1} x^{\frac{3}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.5.1.2

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

$\frac{1}{x^{1 + \frac{3}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.5.2

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

$\frac{1}{x^{\frac{2}{2} + \frac{3}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.5.3

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{2 + 3}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.5.4

Add $2$ and $3$ .

$\frac{1}{x^{\frac{5}{2}} (\frac{13 x^{\frac{3}{2}}}{3} + \frac{2}{3})}$

Step 8.1.6

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{5}{2}} \frac{13 x^{\frac{3}{2}} + 2}{3}}$

Step 8.2

Combine $x^{\frac{5}{2}}$ and $\frac{13 x^{\frac{3}{2}} + 2}{3}$ .

$\frac{1}{\frac{x^{\frac{5}{2}} (13 x^{\frac{3}{2}} + 2)}{3}}$

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{x^{\frac{5}{2}} (13 x^{\frac{3}{2}} + 2)}$

Step 8.4

Multiply $\frac{3}{x^{\frac{5}{2}} (13 x^{\frac{3}{2}} + 2)}$ by $1$ .

$\frac{3}{x^{\frac{5}{2}} (13 x^{\frac{3}{2}} + 2)}$