Calculus Examples

$\int_{0}^{1} 2 x e^{x} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} x e^{x} d x$

Step 2

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

$2 (x e^{x}]_{0}^{1} - \int_{0}^{1} e^{x} d x)$

Step 3

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

$2 (x e^{x}]_{0}^{1} - (e^{x}]_{0}^{1}))$

Step 4

Substitute and simplify.

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Step 4.1

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

$2 ((1 e^{1}) + 0 e^{0} - (e^{x}]_{0}^{1}))$

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify.

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

Step 4.3.2

Multiply $e$ by $1$ .

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

Step 4.3.3

Anything raised to $0$ is $1$ .

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

Step 4.3.4

Multiply $0$ by $1$ .

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

Step 4.3.5

Add $e$ and $0$ .

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

Step 4.3.6

Simplify.

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

Step 4.3.7

Anything raised to $0$ is $1$ .

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

Step 4.3.8

Multiply $- 1$ by $1$ .

$2 (e - (e - 1))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$2 (e - e - - 1)$

Step 5.1.2

Multiply $- 1$ by $- 1$ .

$2 (e - e + 1)$

Step 5.2

Subtract $e$ from $e$ .

$2 (0 + 1)$

Step 5.3

Add $0$ and $1$ .

$2 \cdot 1$

Step 5.4

Multiply $2$ by $1$ .

$2$

Step 6