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

Simplify the answer.

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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

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 6.5

Subtract $1$ from $1$ .

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

Step 6.6

Simplify.

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

Step 6.7

Multiply $0$ by $e$ .

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

Step 6.8

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 6.9

Subtract $1$ from $0$ .

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

Step 6.10

Multiply $- 1$ by $- 1$ .

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

Step 6.11

Anything raised to $0$ is $1$ .

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

Step 6.12

Multiply $1$ by $1$ .

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

Step 6.13

Add $0$ and $1$ .

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

Step 6.14

Simplify.

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

Step 6.15

Multiply $e$ by $1$ .

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

Step 6.16

Anything raised to $0$ is $1$ .

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

Step 6.17

Multiply $0$ by $1$ .

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

Step 6.18

Add $e$ and $0$ .

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

Step 6.19

Simplify.

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

Step 6.20

Anything raised to $0$ is $1$ .

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

Step 6.21

Multiply $- 1$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1$