Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $- 3 t$ .

$\frac{d}{d t} [- 3 t]$

Step 1.1.2

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

$- 3 \frac{d}{d t} [t]$

Step 1.1.3

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

$- 3 \cdot 1$

Step 1.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

$- \frac{1}{3} (e^{u} + C)$

Step 6

Simplify.

$- \frac{1}{3} e^{u} + C$

Step 7

Replace all occurrences of $u$ with $- 3 t$ .

$- \frac{1}{3} e^{- 3 t} + C$