Calculus Examples

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

Step 1

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $- t^{2}$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $2$ by $- 1$ .

$- 2 t$

Step 2.2

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

$3 \int e^{u} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$3 \int e^{u} (- \frac{1}{2}) d u$

Step 3.2

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

$3 \int - \frac{e^{u}}{2} d u$

Step 4

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

$3 (- \int \frac{e^{u}}{2} d u)$

Step 5

Multiply $- 1$ by $3$ .

$- 3 \int \frac{e^{u}}{2} d u$

Step 6

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

$- 3 (\frac{1}{2} \int e^{u} d u)$

Step 7

Simplify.

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Step 7.1

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

$\frac{- 3}{2} \int e^{u} d u$

Step 7.2

Move the negative in front of the fraction.

$- \frac{3}{2} \int e^{u} d u$

Step 8

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

$- \frac{3}{2} (e^{u} + C)$

Step 9

Simplify.

$- \frac{3}{2} e^{u} + C$

Step 10

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

$- \frac{3}{2} e^{- t^{2}} + C$