Calculus Examples

$\int_{0}^{1} x e^{x} d x$

Step 1

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

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

Step 2

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

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

Step 3

Substitute and simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify.

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

Step 3.3.2

Multiply $e$ by $1$ .

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

Step 3.3.3

Anything raised to $0$ is $1$ .

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

Step 3.3.4

Multiply $0$ by $1$ .

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

Step 3.3.5

Add $e$ and $0$ .

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

Step 3.3.6

Simplify.

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

Step 3.3.7

Anything raised to $0$ is $1$ .

$e - (e - 1 \cdot 1)$

Step 3.3.8

Multiply $- 1$ by $1$ .

$e - (e - 1)$

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.

$e - e - - 1$

Step 4.1.2

Multiply $- 1$ by $- 1$ .

$e - e + 1$

Step 4.2

Subtract $e$ from $e$ .

$0 + 1$

Step 4.3

Add $0$ and $1$ .

$1$

Step 5