Calculus Examples

$\int r^{2} e^{- \frac{2 r}{a}} d r$

Step 1

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

$r^{2} (- \frac{1}{2} e^{- \frac{2 r}{a}} a) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a (2 r) d r$

Step 2

Simplify.

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Step 2.1

Combine $e^{- \frac{2 r}{a}}$ and $\frac{1}{2}$ .

$r^{2} (- \frac{e^{- \frac{2 r}{a}}}{2} a) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a (2 r) d r$

Step 2.2

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

$r^{2} (- \frac{a e^{- \frac{2 r}{a}}}{2}) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a (2 r) d r$

Step 2.3

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a (2 r) d r$

Step 3

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (- \frac{1}{2} a \cdot 2 \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4

Simplify.

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Step 4.1

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (- \frac{a}{2} \cdot 2 \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.2

Multiply $2$ by $- 1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (- 2 \frac{a}{2} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.3

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (\frac{- 2 a}{2} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.4

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

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Step 4.4.1

Factor $2$ out of $- 2 a$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (\frac{2 (- a)}{2} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $2$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (\frac{2 (- a)}{2 (1)} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.4.2.2

Cancel the common factor.

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (\frac{2 (- a)}{2 \cdot 1} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.4.2.3

Rewrite the expression.

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (\frac{- a}{1} \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.4.2.4

Divide $- a$ by $1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} - (- a \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.5

Multiply $- 1$ by $- 1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + 1 (a \int e^{- \frac{2 r}{a}} (r) d r)$

Step 4.6

Multiply $a$ by $1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a \int e^{- \frac{2 r}{a}} (r) d r$

Step 5

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (r (- \frac{1}{2} e^{- \frac{2 r}{a}} a) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a d r)$

Step 6

Simplify.

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Step 6.1

Combine $e^{- \frac{2 r}{a}}$ and $\frac{1}{2}$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (r (- \frac{e^{- \frac{2 r}{a}}}{2} a) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a d r)$

Step 6.2

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (r (- \frac{a e^{- \frac{2 r}{a}}}{2}) - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a d r)$

Step 6.3

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \int - \frac{1}{2} e^{- \frac{2 r}{a}} a d r)$

Step 6.4

Combine $e^{- \frac{2 r}{a}}$ and $\frac{1}{2}$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \int - \frac{e^{- \frac{2 r}{a}}}{2} a d r)$

Step 6.5

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \int - \frac{a e^{- \frac{2 r}{a}}}{2} d r)$

Step 7

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - - \int \frac{a e^{- \frac{2 r}{a}}}{2} d r)$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + 1 \int \frac{a e^{- \frac{2 r}{a}}}{2} d r)$

Step 8.2

Multiply $\int \frac{a e^{- \frac{2 r}{a}}}{2} d r$ by $1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \int \frac{a e^{- \frac{2 r}{a}}}{2} d r)$

Step 9

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int e^{- \frac{2 r}{a}} d r)$

Step 10

Let $u = - \frac{2 r}{a}$ . Then $d u = - \frac{2}{a} d r$ , so $- \frac{a}{2} d u = d r$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.1

Let $u = - \frac{2 r}{a}$ . Find $\frac{d u}{d r}$ .

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Step 10.1.1

Differentiate $- \frac{2 r}{a}$ .

$\frac{d}{d r} [- \frac{2 r}{a}]$

Step 10.1.2

Since $- \frac{2}{a}$ is constant with respect to $r$ , the derivative of $- \frac{2 r}{a}$ with respect to $r$ is $- \frac{2}{a} \frac{d}{d r} [r]$ .

$- \frac{2}{a} \frac{d}{d r} [r]$

Step 10.1.3

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

$- \frac{2}{a} \cdot 1$

Step 10.1.4

Multiply $- 1$ by $1$ .

$- \frac{2}{a}$

Step 10.2

Rewrite the problem using $u$ and $d u$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int - e^{u} \frac{- 1}{- \frac{2}{a}} d u)$

Step 11

Simplify.

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Step 11.1

Dividing two negative values results in a positive value.

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int - e^{u} \frac{1}{\frac{2}{a}} d u)$

Step 11.2

Multiply by the reciprocal of the fraction to divide by $\frac{2}{a}$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int - e^{u} (1 \frac{a}{2}) d u)$

Step 11.3

Multiply $\frac{a}{2}$ by $1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int - e^{u} \frac{a}{2} d u)$

Step 11.4

Combine $\frac{a}{2}$ and $e^{u}$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} \int - \frac{a e^{u}}{2} d u)$

Step 12

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} (- \int \frac{a e^{u}}{2} d u))$

Step 13

Since $\frac{a}{2}$ is constant with respect to $u$ , move $\frac{a}{2}$ out of the integral.

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} + \frac{a}{2} (- (\frac{a}{2} \int e^{u} d u)))$

Step 14

Simplify.

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Step 14.1

Multiply $\frac{a}{2}$ by $\frac{a}{2}$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a \cdot a}{2 \cdot 2} \int e^{u} d u)$

Step 14.2

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{1} a}{2 \cdot 2} \int e^{u} d u)$

Step 14.3

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{1} a^{1}}{2 \cdot 2} \int e^{u} d u)$

Step 14.4

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{1 + 1}}{2 \cdot 2} \int e^{u} d u)$

Step 14.5

Add $1$ and $1$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{2}}{2 \cdot 2} \int e^{u} d u)$

Step 14.6

Multiply $2$ by $2$ .

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{2}}{4} \int e^{u} d u)$

Step 15

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

$- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{2}}{4} (e^{u} + C))$

Step 16

Simplify.

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Step 16.1

Rewrite $- \frac{r^{2} (a e^{- \frac{2 r}{a}})}{2} + a (- \frac{r (a e^{- \frac{2 r}{a}})}{2} - \frac{a^{2}}{4} (e^{u} + C))$ as $- \frac{1}{2} r^{2} a e^{- \frac{2 r}{a}} + a (- \frac{1}{2} r a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$ .

$- \frac{1}{2} r^{2} a e^{- \frac{2 r}{a}} + a (- \frac{1}{2} r a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2

Simplify.

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Step 16.2.1

Combine $r^{2}$ and $\frac{1}{2}$ .

$- \frac{r^{2}}{2} a e^{- \frac{2 r}{a}} + a (- \frac{1}{2} r a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.2

Combine $a$ and $\frac{r^{2}}{2}$ .

$- \frac{a r^{2}}{2} e^{- \frac{2 r}{a}} + a (- \frac{1}{2} r a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.3

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

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{1}{2} r a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.4

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

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{r}{2} a e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.5

Combine $a$ and $\frac{r}{2}$ .

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{a r}{2} e^{- \frac{2 r}{a}} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.6

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

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{1}{4} a^{2} e^{u}) + C$

Step 16.2.7

Combine $a^{2}$ and $\frac{1}{4}$ .

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{a^{2}}{4} e^{u}) + C$

Step 16.2.8

Combine $e^{u}$ and $\frac{a^{2}}{4}$ .

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4}) + C$

Step 16.2.9

To write $a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4}) \cdot \frac{2}{2} + C$

Step 16.2.10

Combine $a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4})$ and $\frac{2}{2}$ .

$- \frac{e^{- \frac{2 r}{a}} (a r^{2})}{2} + \frac{a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4}) \cdot 2}{2} + C$

Step 16.2.11

Combine the numerators over the common denominator.

$\frac{- e^{- \frac{2 r}{a}} (a r^{2}) + a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4}) \cdot 2}{2} + C$

Step 16.2.12

Move $2$ to the left of $a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4})$ .

$\frac{- e^{- \frac{2 r}{a}} (a r^{2}) + 2 a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{u} a^{2}}{4})}{2} + C$

Step 17

Replace all occurrences of $u$ with $- \frac{2 r}{a}$ .

$\frac{- e^{- \frac{2 r}{a}} (a r^{2}) + 2 a (- \frac{e^{- \frac{2 r}{a}} (a r)}{2} - \frac{e^{- \frac{2 r}{a}} a^{2}}{4})}{2} + C$

Step 18

Reorder terms.

$\frac{1}{2} (- e^{- \frac{2 r}{a}} (a r^{2}) + 2 a (- \frac{1}{2} e^{- \frac{2 r}{a}} a r - \frac{1}{4} e^{- \frac{2 r}{a}} a^{2})) + C$