Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2} - 1$ and $d v = e^{x}$ .

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

Step 2

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

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

Step 3

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $(x^{2} - 1) e^{x}$ at $1$ and at $0$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Simplify.

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Step 6.4.1

One to any power is one.

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

Step 6.4.2

Subtract $1$ from $1$ .

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

Step 6.4.3

Simplify.

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

Step 6.4.4

Multiply $0$ by $e$ .

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

Step 6.4.5

Raising $0$ to any positive power yields $0$ .

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

Step 6.4.6

Subtract $1$ from $0$ .

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

Step 6.4.7

Multiply $- 1$ by $- 1$ .

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

Step 6.4.8

Anything raised to $0$ is $1$ .

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

Step 6.4.9

Multiply $1$ by $1$ .

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

Step 6.4.10

Add $0$ and $1$ .

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

Step 6.4.11

Simplify.

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

Step 6.4.12

Multiply $e$ by $1$ .

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

Step 6.4.13

Anything raised to $0$ is $1$ .

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

Step 6.4.14

Multiply $0$ by $1$ .

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

Step 6.4.15

Add $e$ and $0$ .

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

Step 6.4.16

Simplify.

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

Step 6.4.17

Anything raised to $0$ is $1$ .

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

Step 6.4.18

Multiply $- 1$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Apply the distributive property.

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

Step 7.1.1.2

Multiply $- 1$ by $- 1$ .

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

Step 7.1.2

Subtract $e$ from $e$ .

$1 - 2 (0 + 1)$

Step 7.1.3

Add $0$ and $1$ .

$1 - 2 \cdot 1$

Step 7.1.4

Multiply $- 2$ by $1$ .

$1 - 2$

Step 7.2

Subtract $2$ from $1$ .

$- 1$

Step 8