Calculus Examples

$\int_{0}^{1} (x - 3) e^{x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x - 3$ and $d v = e^{x}$ .

$(x - 3) e^{x}]_{0}^{1} - \int_{0}^{1} e^{x} d x$

Step 2

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

$(x - 3) e^{x}]_{0}^{1} - (e^{x}]_{0}^{1})$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $(x - 3) e^{x}$ at $1$ and at $0$ .

$((1 - 3) e^{1}) - (0 - 3) e^{0} - (e^{x}]_{0}^{1})$

Step 3.2

Evaluate $e^{x}$ at $1$ and at $0$ .

$((1 - 3) e^{1}) - (0 - 3) e^{0} - ((e^{1}) - e^{0})$

Step 3.3

Simplify.

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Step 3.3.1

Subtract $3$ from $1$ .

$- 2 e^{1} - (0 - 3) e^{0} - ((e^{1}) - e^{0})$

Step 3.3.2

Simplify.

$- 2 e - (0 - 3) e^{0} - ((e^{1}) - e^{0})$

Step 3.3.3

Subtract $3$ from $0$ .

$- 2 e - - 3 e^{0} - ((e^{1}) - e^{0})$

Step 3.3.4

Multiply $- 1$ by $- 3$ .

$- 2 e + 3 e^{0} - ((e^{1}) - e^{0})$

Step 3.3.5

Anything raised to $0$ is $1$ .

$- 2 e + 3 \cdot 1 - ((e^{1}) - e^{0})$

Step 3.3.6

Multiply $3$ by $1$ .

$- 2 e + 3 - ((e^{1}) - e^{0})$

Step 3.3.7

Simplify.

$- 2 e + 3 - (e - e^{0})$

Step 3.3.8

Anything raised to $0$ is $1$ .

$- 2 e + 3 - (e - 1 \cdot 1)$

Step 3.3.9

Multiply $- 1$ by $1$ .

$- 2 e + 3 - (e - 1)$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

$- 2 e + 3 - e - - 1$

Step 4.1.2

Multiply $- 1$ by $- 1$ .

$- 2 e + 3 - e + 1$

Step 4.2

Subtract $e$ from $- 2 e$ .

$3 - 3 e + 1$

Step 4.3

Add $3$ and $1$ .

$4 - 3 e$

Step 5

The result can be shown in multiple forms.

Exact Form:

$4 - 3 e$

Decimal Form:

$- 4.15484548 \dots$

Step 6