Calculus Examples

$(1 - t) e^{2 t - t^{2}}$

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $2 t - t^{2}$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $2 t - t^{2}$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [- t^{2}]$ .

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

Step 1.1.3

Evaluate $\frac{d}{d t} [2 t]$ .

Tap for more steps...

Step 1.1.3.1

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

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

Step 1.1.3.2

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

$2 \cdot 1 + \frac{d}{d t} [- t^{2}]$

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

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

Tap for more steps...

Step 1.1.4.1

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

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

Step 1.1.4.2

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

$2 - (2 t)$

Step 1.1.4.3

Multiply $2$ by $- 1$ .

$2 - 2 t$

Step 1.1.5

Reorder terms.

$- 2 t + 2$

Step 1.1.6

Rearrange terms.

$2 - 2 t$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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

Step 6

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

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