Calculus Examples

$\int_{1}^{2} e^{1 - x} d x$

Step 1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 - x$ .

$\frac{d}{d x} [1 - x]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x]$ .

Tap for more steps...

Step 1.1.3.1

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

$0 - \frac{d}{d x} [x]$

Step 1.1.3.2

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

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 1 \cdot 1$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Multiply $- 1$ by $1$ .

$u_{lower} = 1 - 1$

Step 1.3.2

Subtract $1$ from $1$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

$u_{upper} = 1 - 1 \cdot 2$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $2$ .

$u_{upper} = 1 - 2$

Step 1.5.2

Subtract $2$ from $1$ .

$u_{upper} = - 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 1$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{- 1} - e^{u} d u$

Step 2

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

$- \int_{0}^{- 1} e^{u} d u$

Step 3

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

$- (e^{u}]_{0}^{- 1})$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Evaluate $e^{u}$ at $- 1$ and at $0$ .

$- ((e^{- 1}) - e^{0})$

Step 4.2

Anything raised to $0$ is $1$ .

$- (e^{- 1} - 1 \cdot 1)$

Step 4.3

Multiply $- 1$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.63212055 \dots$