Calculus Examples

$\int_{0}^{2} r^{2} e^{3 r} d r$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = r^{2}$ and $d v = e^{3 r}$ .

$r^{2} (\frac{1}{3} e^{3 r})]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} (2 r) d r$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $e^{3 r}$ .

$r^{2} \frac{e^{3 r}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} (2 r) d r$

Step 2.2

Combine $r^{2}$ and $\frac{e^{3 r}}{3}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} (2 r) d r$

Step 3

Since $\frac{1}{3} \cdot 2$ is constant with respect to $r$ , move $\frac{1}{3} \cdot 2$ out of the integral.

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - (\frac{1}{3} \cdot 2 \int_{0}^{2} e^{3 r} (r) d r)$

Step 4

Combine $\frac{1}{3}$ and $2$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} \int_{0}^{2} e^{3 r} (r) d r$

Step 5

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

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (r (\frac{1}{3} e^{3 r})]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} d r)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $e^{3 r}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (r \frac{e^{3 r}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} d r)$

Step 6.2

Combine $r$ and $\frac{e^{3 r}}{3}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \int_{0}^{2} \frac{1}{3} e^{3 r} d r)$

Step 6.3

Combine $\frac{1}{3}$ and $e^{3 r}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \int_{0}^{2} \frac{e^{3 r}}{3} d r)$

Step 7

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

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - (\frac{1}{3} \int_{0}^{2} e^{3 r} d r))$

Step 8

Let $u = 3 r$ . Then $d u = 3 d r$ , so $\frac{1}{3} d u = d r$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 3 r$ . Find $\frac{d u}{d r}$ .

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Step 8.1.1

Differentiate $3 r$ .

$\frac{d}{d r} [3 r]$

Step 8.1.2

Since $3$ is constant with respect to $r$ , the derivative of $3 r$ with respect to $r$ is $3 \frac{d}{d r} [r]$ .

$3 \frac{d}{d r} [r]$

Step 8.1.3

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

$3 \cdot 1$

Step 8.1.4

Multiply $3$ by $1$ .

$3$

Step 8.2

Substitute the lower limit in for $r$ in $u = 3 r$ .

$u_{lower} = 3 \cdot 0$

Step 8.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 8.4

Substitute the upper limit in for $r$ in $u = 3 r$ .

$u_{upper} = 3 \cdot 2$

Step 8.5

Multiply $3$ by $2$ .

$u_{upper} = 6$

Step 8.6

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

$u_{lower} = 0$

$u_{upper} = 6$

Step 8.7

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

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{3} \int_{0}^{6} e^{u} \frac{1}{3} d u)$

Step 9

Combine $e^{u}$ and $\frac{1}{3}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{3} \int_{0}^{6} \frac{e^{u}}{3} d u)$

Step 10

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

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{3} (\frac{1}{3} \int_{0}^{6} e^{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{3 \cdot 3} \int_{0}^{6} e^{u} d u)$

Step 11.2

Multiply $3$ by $3$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{9} \int_{0}^{6} e^{u} d u)$

Step 12

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{1}{9} e^{u}]_{0}^{6})$

Step 13

Combine $e^{u}]_{0}^{6}$ and $\frac{1}{9}$ .

$\frac{r^{2} e^{3 r}}{3}]_{0}^{2} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{e^{u}]_{0}^{6}}{9})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\frac{r^{2} e^{3 r}}{3}$ at $2$ and at $0$ .

$(\frac{2^{2} e^{3 \cdot 2}}{3}) - \frac{0^{2} e^{3 \cdot 0}}{3} - \frac{2}{3} (\frac{r e^{3 r}}{3}]_{0}^{2} - \frac{e^{u}]_{0}^{6}}{9})$

Step 14.2

Evaluate $\frac{r e^{3 r}}{3}$ at $2$ and at $0$ .

$(\frac{2^{2} e^{3 \cdot 2}}{3}) - \frac{0^{2} e^{3 \cdot 0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{e^{u}]_{0}^{6}}{9})$

Step 14.3

Evaluate $e^{u}$ at $6$ and at $0$ .

$(\frac{2^{2} e^{3 \cdot 2}}{3}) - \frac{0^{2} e^{3 \cdot 0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4

Simplify.

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Step 14.4.1

Raise $2$ to the power of $2$ .

$\frac{4 e^{3 \cdot 2}}{3} - \frac{0^{2} e^{3 \cdot 0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.2

Multiply $3$ by $2$ .

$\frac{4 e^{6}}{3} - \frac{0^{2} e^{3 \cdot 0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.3

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

$\frac{4 e^{6}}{3} - \frac{0 e^{3 \cdot 0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.4

Multiply $3$ by $0$ .

$\frac{4 e^{6}}{3} - \frac{0 e^{0}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.5

Anything raised to $0$ is $1$ .

$\frac{4 e^{6}}{3} - \frac{0 \cdot 1}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.6

Multiply $0$ by $1$ .

$\frac{4 e^{6}}{3} - \frac{0}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.7

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

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Step 14.4.7.1

Factor $3$ out of $0$ .

$\frac{4 e^{6}}{3} - \frac{3 (0)}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.7.2

Cancel the common factors.

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Step 14.4.7.2.1

Factor $3$ out of $3$ .

$\frac{4 e^{6}}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.7.2.2

Cancel the common factor.

$\frac{4 e^{6}}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.7.2.3

Rewrite the expression.

$\frac{4 e^{6}}{3} - \frac{0}{1} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.7.2.4

Divide $0$ by $1$ .

$\frac{4 e^{6}}{3} - 0 - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.8

Multiply $- 1$ by $0$ .

$\frac{4 e^{6}}{3} + 0 - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.9

Add $\frac{4 e^{6}}{3}$ and $0$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} ((\frac{2 e^{3 \cdot 2}}{3}) - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.10

Multiply $3$ by $2$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{0 e^{3 \cdot 0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.11

Multiply $3$ by $0$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{0 e^{0}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.12

Anything raised to $0$ is $1$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{0 \cdot 1}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.13

Multiply $0$ by $1$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{0}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.14

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

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Step 14.4.14.1

Factor $3$ out of $0$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{3 (0)}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.14.2

Cancel the common factors.

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Step 14.4.14.2.1

Factor $3$ out of $3$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.14.2.2

Cancel the common factor.

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.14.2.3

Rewrite the expression.

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{0}{1} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.14.2.4

Divide $0$ by $1$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - 0 - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.15

Multiply $- 1$ by $0$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} + 0 - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.16

Add $\frac{2 e^{6}}{3}$ and $0$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{(e^{6}) - e^{0}}{9})$

Step 14.4.17

Anything raised to $0$ is $1$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{e^{6} - 1 \cdot 1}{9})$

Step 14.4.18

Multiply $- 1$ by $1$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} - \frac{e^{6} - 1}{9})$

Step 14.4.19

To write $\frac{2 e^{6}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6}}{3} \cdot \frac{3}{3} - \frac{e^{6} - 1}{9})$

Step 14.4.20

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.4.20.1

Multiply $\frac{2 e^{6}}{3}$ by $\frac{3}{3}$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6} \cdot 3}{3 \cdot 3} - \frac{e^{6} - 1}{9})$

Step 14.4.20.2

Multiply $3$ by $3$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} (\frac{2 e^{6} \cdot 3}{9} - \frac{e^{6} - 1}{9})$

Step 14.4.21

Combine the numerators over the common denominator.

$\frac{4 e^{6}}{3} - \frac{2}{3} \cdot \frac{2 e^{6} \cdot 3 - (e^{6} - 1)}{9}$

Step 14.4.22

Multiply $3$ by $2$ .

$\frac{4 e^{6}}{3} - \frac{2}{3} \cdot \frac{6 e^{6} - (e^{6} - 1)}{9}$

Step 14.4.23

Multiply $\frac{6 e^{6} - (e^{6} - 1)}{9}$ by $\frac{2}{3}$ .

$\frac{4 e^{6}}{3} - \frac{(6 e^{6} - (e^{6} - 1)) \cdot 2}{9 \cdot 3}$

Step 14.4.24

Multiply $9$ by $3$ .

$\frac{4 e^{6}}{3} - \frac{(6 e^{6} - (e^{6} - 1)) \cdot 2}{27}$

Step 14.4.25

Move $2$ to the left of $6 e^{6} - (e^{6} - 1)$ .

$\frac{4 e^{6}}{3} - \frac{2 \cdot (6 e^{6} - (e^{6} - 1))}{27}$

Step 14.4.26

To write $\frac{4 e^{6}}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4 e^{6}}{3} \cdot \frac{9}{9} - \frac{2 (6 e^{6} - (e^{6} - 1))}{27}$

Step 14.4.27

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.4.27.1

Multiply $\frac{4 e^{6}}{3}$ by $\frac{9}{9}$ .

$\frac{4 e^{6} \cdot 9}{3 \cdot 9} - \frac{2 (6 e^{6} - (e^{6} - 1))}{27}$

Step 14.4.27.2

Multiply $3$ by $9$ .

$\frac{4 e^{6} \cdot 9}{27} - \frac{2 (6 e^{6} - (e^{6} - 1))}{27}$

Step 14.4.28

Combine the numerators over the common denominator.

$\frac{4 e^{6} \cdot 9 - 2 (6 e^{6} - (e^{6} - 1))}{27}$

Step 14.4.29

Multiply $9$ by $4$ .

$\frac{36 e^{6} - 2 (6 e^{6} - (e^{6} - 1))}{27}$

Step 15

Simplify the numerator.

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Step 15.1

Simplify each term.

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Step 15.1.1

Apply the distributive property.

$\frac{36 e^{6} - 2 (6 e^{6} - e^{6} - - 1)}{27}$

Step 15.1.2

Multiply $- 1$ by $- 1$ .

$\frac{36 e^{6} - 2 (6 e^{6} - e^{6} + 1)}{27}$

Step 15.2

Subtract $e^{6}$ from $6 e^{6}$ .

$\frac{36 e^{6} - 2 (5 e^{6} + 1)}{27}$

Step 15.3

Apply the distributive property.

$\frac{36 e^{6} - 2 (5 e^{6}) - 2 \cdot 1}{27}$

Step 15.4

Multiply $5$ by $- 2$ .

$\frac{36 e^{6} - 10 e^{6} - 2 \cdot 1}{27}$

Step 15.5

Multiply $- 2$ by $1$ .

$\frac{36 e^{6} - 10 e^{6} - 2}{27}$

Step 15.6

Subtract $10 e^{6}$ from $36 e^{6}$ .

$\frac{26 e^{6} - 2}{27}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{26 e^{6} - 2}{27}$

Decimal Form:

$388.41291225 \dots$

Step 17