Calculus Examples

$\int t^{3} e^{- t^{2}} d t$

Step 1

Let $u = - t^{2}$ . Then $d u = - 2 t d t$ , so $- \frac{1}{2} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = - t^{2}$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 1.1.1

Differentiate $- t^{2}$ .

$\frac{d}{d t} [- t^{2}]$

Step 1.1.2

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

$- \frac{d}{d t} [t^{2}]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$- (2 t)$

Step 1.1.4

Multiply $2$ by $- 1$ .

$- 2 t$

Step 1.2

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

$\int {\sqrt{- u}}^{2} e^{u} \frac{1}{- 2} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Rewrite ${\sqrt{- u}}^{2}$ as $- u$ .

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Simplify.

$\int - u e^{u} \frac{1}{- 2} d u$

Step 2.2

Move the negative in front of the fraction.

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

Step 2.3

Multiply $- 1$ by $- 1$ .

$\int 1 u e^{u} \frac{1}{2} d u$

Step 2.4

Multiply $u$ by $1$ .

$\int u e^{u} \frac{1}{2} d u$

Step 2.5

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

$\int \frac{u}{2} e^{u} d u$

Step 2.6

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

$\int \frac{u e^{u}}{2} d u$

Step 3

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

$\frac{1}{2} \int u e^{u} d u$

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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

Step 7

Replace all occurrences of $u$ with $- t^{2}$ .

$\frac{1}{2} (- t^{2} e^{- t^{2}} - e^{- t^{2}}) + C$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Apply the distributive property.

$\frac{1}{2} (- t^{2} e^{- t^{2}}) + \frac{1}{2} (- e^{- t^{2}}) + C$

Step 8.2

Multiply $\frac{1}{2} (- t^{2} e^{- t^{2}})$ .

Tap for more steps...

Step 8.2.1

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

$- \frac{t^{2}}{2} e^{- t^{2}} + \frac{1}{2} (- e^{- t^{2}}) + C$

Step 8.2.2

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

$- \frac{e^{- t^{2}} t^{2}}{2} + \frac{1}{2} (- e^{- t^{2}}) + C$

Step 8.3

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

$- \frac{e^{- t^{2}} t^{2}}{2} - \frac{e^{- t^{2}}}{2} + C$

Step 8.4

Reorder factors in $- \frac{e^{- t^{2}} t^{2}}{2} - \frac{e^{- t^{2}}}{2}$ .

$- \frac{t^{2} e^{- t^{2}}}{2} - \frac{e^{- t^{2}}}{2} + C$

Step 9

Reorder terms.

$- \frac{1}{2} t^{2} e^{- t^{2}} - \frac{1}{2} e^{- t^{2}} + C$