Calculus Examples

$\int_{0}^{3} 10 x (3^{- x}) d x$

Step 1

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int_{0}^{3} x \cdot (3^{- x}) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = 3^{- x}$ .

$10 (x (- \frac{1}{\ln (3)} \cdot 3^{- x})]_{0}^{3} - \int_{0}^{3} - \frac{1}{\ln (3)} \cdot 3^{- x} d x)$

Step 3

Simplify.

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Step 3.1

Combine $3^{- x}$ and $\frac{1}{\ln (3)}$ .

$10 (x (- \frac{3^{- x}}{\ln (3)})]_{0}^{3} - \int_{0}^{3} - \frac{1}{\ln (3)} \cdot 3^{- x} d x)$

Step 3.2

Combine $x$ and $\frac{3^{- x}}{\ln (3)}$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} - \int_{0}^{3} - \frac{1}{\ln (3)} \cdot 3^{- x} d x)$

Step 3.3

Combine $3^{- x}$ and $\frac{1}{\ln (3)}$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} - \int_{0}^{3} - \frac{3^{- x}}{\ln (3)} d x)$

Step 4

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

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} - - \int_{0}^{3} \frac{3^{- x}}{\ln (3)} d x)$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + 1 \int_{0}^{3} \frac{3^{- x}}{\ln (3)} d x)$

Step 5.2

Multiply $\int_{0}^{3} \frac{3^{- x}}{\ln (3)} d x$ by $1$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + \int_{0}^{3} \frac{3^{- x}}{\ln (3)} d x)$

Step 6

Since $\frac{1}{\ln (3)}$ is constant with respect to $x$ , move $\frac{1}{\ln (3)}$ out of the integral.

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + \frac{1}{\ln (3)} \int_{0}^{3} 3^{- x} d x)$

Step 7

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 7.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 7.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.2

Substitute the lower limit in for $x$ in $u = - x$ .

$u_{lower} = - 0$

Step 7.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 7.4

Substitute the upper limit in for $x$ in $u = - x$ .

$u_{upper} = - 1 \cdot 3$

Step 7.5

Multiply $- 1$ by $3$ .

$u_{upper} = - 3$

Step 7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 3$

Step 7.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + \frac{1}{\ln (3)} \int_{0}^{- 3} - 3^{u} d u)$

Step 8

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

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + \frac{1}{\ln (3)} (- \int_{0}^{- 3} 3^{u} d u))$

Step 9

The integral of $3^{u}$ with respect to $u$ is $\frac{3^{u}}{\ln (3)}$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} + \frac{1}{\ln (3)} (- (\frac{3^{u}}{\ln (3)}]_{0}^{- 3})))$

Step 10

Combine $\frac{1}{\ln (3)}$ and $\frac{3^{u}}{\ln (3)}]_{0}^{- 3}$ .

$10 (- \frac{x \cdot 3^{- x}}{\ln (3)}]_{0}^{3} - \frac{\frac{3^{u}}{\ln (3)}]_{0}^{- 3}}{\ln (3)})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $- \frac{x \cdot 3^{- x}}{\ln (3)}$ at $3$ and at $0$ .

$10 ((- \frac{3 \cdot 3^{- 1 \cdot 3}}{\ln (3)}) + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{\frac{3^{u}}{\ln (3)}]_{0}^{- 3}}{\ln (3)})$

Step 11.2

Evaluate $\frac{3^{u}}{\ln (3)}$ at $- 3$ and at $0$ .

$10 ((- \frac{3 \cdot 3^{- 1 \cdot 3}}{\ln (3)}) + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $- 1$ by $3$ .

$10 (- \frac{3 \cdot 3^{- 3}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 (- \frac{3 \cdot \frac{1}{3^{3}}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.3

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

$10 (- \frac{3 \cdot \frac{1}{27}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.4

Combine $3$ and $\frac{1}{27}$ .

$10 (- \frac{\frac{3}{27}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.5

Cancel the common factor of $3$ and $27$ .

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Step 11.3.5.1

Factor $3$ out of $3$ .

$10 (- \frac{\frac{3 (1)}{27}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.5.2

Cancel the common factors.

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Step 11.3.5.2.1

Factor $3$ out of $27$ .

$10 (- \frac{\frac{3 \cdot 1}{3 \cdot 9}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.5.2.2

Cancel the common factor.

$10 (- \frac{\frac{3 \cdot 1}{3 \cdot 9}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.5.2.3

Rewrite the expression.

$10 (- \frac{\frac{1}{9}}{\ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.6

Rewrite $\frac{\frac{1}{9}}{\ln (3)}$ as a product.

$10 (- (\frac{1}{9} \cdot \frac{1}{\ln (3)}) + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.7

Multiply $\frac{1}{9}$ by $\frac{1}{\ln (3)}$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0 \cdot 3^{- 0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.8

Multiply $- 1$ by $0$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0 \cdot 3^{0}}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.9

Anything raised to $0$ is $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0 \cdot 1}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.10

Multiply $0$ by $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{(\frac{3^{- 3}}{\ln (3)}) - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.11

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{\frac{1}{3^{3}}}{\ln (3)} - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.12

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

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{\frac{1}{27}}{\ln (3)} - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.13

Rewrite $\frac{\frac{1}{27}}{\ln (3)}$ as a product.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{1}{27} \cdot \frac{1}{\ln (3)} - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.14

Multiply $\frac{1}{27}$ by $\frac{1}{\ln (3)}$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{1}{27 \ln (3)} - \frac{3^{0}}{\ln (3)}}{\ln (3)})$

Step 11.3.15

Anything raised to $0$ is $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{1}{27 \ln (3)} - \frac{1}{\ln (3)}}{\ln (3)})$

Step 11.3.16

Multiply $\frac{\frac{1}{27 \ln (3)} - \frac{1}{\ln (3)}}{\ln (3)}$ by $\frac{27 \ln (3)}{27 \ln (3)}$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - (\frac{27 \ln (3)}{27 \ln (3)} \cdot \frac{\frac{1}{27 \ln (3)} - \frac{1}{\ln (3)}}{\ln (3)}))$

Step 11.3.17

Combine.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{27 \ln (3) (\frac{1}{27 \ln (3)} - \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.18

Apply the distributive property.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{27 \ln (3) \frac{1}{27 \ln (3)} + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.19

Combine $\frac{1}{27 \ln (3)}$ and $27$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{27}{27 \ln (3)} \ln (3) + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.20

Combine $\frac{27}{27 \ln (3)}$ and $\ln (3)$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{27 \ln (3)}{27 \ln (3)} + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.21

Cancel the common factor of $27$ .

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Step 11.3.21.1

Cancel the common factor.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{27 \ln (3)}{27 \ln (3)} + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.21.2

Rewrite the expression.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{\ln (3)}{\ln (3)} + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.22

Cancel the common factor of $\ln (3)$ .

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Step 11.3.22.1

Cancel the common factor.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{\frac{\ln (3)}{\ln (3)} + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.22.2

Rewrite the expression.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 + 27 \ln (3) (- \frac{1}{\ln (3)})}{27 \ln (3) \ln (3)})$

Step 11.3.23

Multiply $- 1$ by $27$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 - 27 \ln (3) \frac{1}{\ln (3)}}{27 \ln (3) \ln (3)})$

Step 11.3.24

Combine $\frac{1}{\ln (3)}$ and $- 27$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 + \frac{- 27}{\ln (3)} \ln (3)}{27 \ln (3) \ln (3)})$

Step 11.3.25

Combine $\frac{- 27}{\ln (3)}$ and $\ln (3)$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 + \frac{- 27 \ln (3)}{\ln (3)}}{27 \ln (3) \ln (3)})$

Step 11.3.26

Cancel the common factor of $\ln (3)$ .

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Step 11.3.26.1

Cancel the common factor.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 + \frac{- 27 \ln (3)}{\ln (3)}}{27 \ln (3) \ln (3)})$

Step 11.3.26.2

Divide $- 27$ by $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{1 - 27}{27 \ln (3) \ln (3)})$

Step 11.3.27

Subtract $27$ from $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{- 26}{27 \ln (3) \ln (3)})$

Step 11.3.28

Raise $\ln (3)$ to the power of $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{- 26}{27 (\ln^{1} (3) \ln (3))})$

Step 11.3.29

Raise $\ln (3)$ to the power of $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{- 26}{27 (\ln^{1} (3) \ln^{1} (3))})$

Step 11.3.30

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

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{- 26}{27 {\ln (3)}^{1 + 1}})$

Step 11.3.31

Add $1$ and $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - \frac{- 26}{27 \ln^{2} (3)})$

Step 11.3.32

Move the negative in front of the fraction.

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} - - \frac{26}{27 \ln^{2} (3)})$

Step 11.3.33

Multiply $- 1$ by $- 1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} + 1 \frac{26}{27 \ln^{2} (3)})$

Step 11.3.34

Multiply $\frac{26}{27 \ln^{2} (3)}$ by $1$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{0}{\ln (3)} + \frac{26}{27 \ln^{2} (3)})$

Step 12

Simplify.

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Step 12.1

Divide $0$ by $\ln (3)$ .

$10 (- \frac{1}{9 \ln (3)} + 0 + \frac{26}{27 \ln^{2} (3)})$

Step 12.2

Add $- \frac{1}{9 \ln (3)}$ and $0$ .

$10 (- \frac{1}{9 \ln (3)} + \frac{26}{27 \ln^{2} (3)})$

Step 12.3

Apply the distributive property.

$10 (- \frac{1}{9 \ln (3)}) + 10 \frac{26}{27 \ln^{2} (3)}$

Step 12.4

Multiply $10 (- \frac{1}{9 \ln (3)})$ .

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Step 12.4.1

Multiply $- 1$ by $10$ .

$- 10 \frac{1}{9 \ln (3)} + 10 \frac{26}{27 \ln^{2} (3)}$

Step 12.4.2

Combine $- 10$ and $\frac{1}{9 \ln (3)}$ .

$\frac{- 10}{9 \ln (3)} + 10 \frac{26}{27 \ln^{2} (3)}$

Step 12.5

Multiply $10 \frac{26}{27 \ln^{2} (3)}$ .

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Step 12.5.1

Combine $10$ and $\frac{26}{27 \ln^{2} (3)}$ .

$\frac{- 10}{9 \ln (3)} + \frac{10 \cdot 26}{27 \ln^{2} (3)}$

Step 12.5.2

Multiply $10$ by $26$ .

$\frac{- 10}{9 \ln (3)} + \frac{260}{27 \ln^{2} (3)}$

Step 12.6

Move the negative in front of the fraction.

$- \frac{10}{9 \ln (3)} + \frac{260}{27 \ln^{2} (3)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{10}{9 \ln (3)} + \frac{260}{27 \ln^{2} (3)}$

Decimal Form:

$6.96711259 \dots$

Step 14