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

Multiply $2$ by $- 1$ .

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

Step 3

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 4

Multiply $- 2$ by $- 1$ .

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

Step 5

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 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

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

Step 7.2

Multiply $\int_{0}^{1} e^{- x} d x$ by $1$ .

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

Step 8

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 8.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 8.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 8.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 8.2

Substitute the lower limit in for $x$ in $u = - x$ .

$u_{lower} = - 0$

Step 8.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 8.4

Substitute the upper limit in for $x$ in $u = - x$ .

$u_{upper} = - 1 \cdot 1$

Step 8.5

Multiply $- 1$ by $1$ .

$u_{upper} = - 1$

Step 8.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 1$

Step 8.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

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

Step 9

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 10

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

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

Step 11

Substitute and simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

Evaluate $e^{u}$ at $- 1$ and at $0$ .

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

Step 11.4

Simplify.

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Step 11.4.1

One to any power is one.

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

Step 11.4.2

Add $1$ and $1$ .

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

Step 11.4.3

Multiply $- 1$ by $1$ .

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

Step 11.4.4

Multiply $- 1$ by $2$ .

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

Step 11.4.5

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

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

Step 11.4.6

Add $0$ and $1$ .

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

Step 11.4.7

Multiply $- 1$ by $1$ .

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

Step 11.4.8

Multiply $- 1$ by $0$ .

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

Step 11.4.9

Anything raised to $0$ is $1$ .

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

Step 11.4.10

Multiply $- 1$ by $1$ .

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

Step 11.4.11

Multiply $- 1$ by $- 1$ .

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

Step 11.4.12

Multiply $- 1$ by $1$ .

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

Step 11.4.13

Multiply $- 1$ by $1$ .

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

Step 11.4.14

Multiply $- 1$ by $0$ .

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

Step 11.4.15

Anything raised to $0$ is $1$ .

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

Step 11.4.16

Multiply $- 1$ by $1$ .

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

Step 11.4.17

Multiply $0$ by $- 1$ .

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

Step 11.4.18

Add $- e^{- 1}$ and $0$ .

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

Step 11.4.19

Anything raised to $0$ is $1$ .

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

Step 11.4.20

Multiply $- 1$ by $1$ .

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

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12.1.2

Combine $- 2$ and $\frac{1}{e}$ .

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

Step 12.1.3

Move the negative in front of the fraction.

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

Step 12.1.4

Simplify each term.

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Step 12.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12.1.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{e} + 1 + 2 (- \frac{1}{e} - (\frac{1}{e} - 1))$

Step 12.1.4.3

Apply the distributive property.

$- \frac{2}{e} + 1 + 2 (- \frac{1}{e} - \frac{1}{e} - - 1)$

Step 12.1.4.4

Multiply $- 1$ by $- 1$ .

$- \frac{2}{e} + 1 + 2 (- \frac{1}{e} - \frac{1}{e} + 1)$

Step 12.1.5

Combine the numerators over the common denominator.

$- \frac{2}{e} + 1 + 2 (1 + \frac{- 1 - 1}{e})$

Step 12.1.6

Subtract $1$ from $- 1$ .

$- \frac{2}{e} + 1 + 2 (1 + \frac{- 2}{e})$

Step 12.1.7

Move the negative in front of the fraction.

$- \frac{2}{e} + 1 + 2 (1 - \frac{2}{e})$

Step 12.1.8

Apply the distributive property.

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

Step 12.1.9

Multiply $2$ by $1$ .

$- \frac{2}{e} + 1 + 2 + 2 (- \frac{2}{e})$

Step 12.1.10

Multiply $2 (- \frac{2}{e})$ .

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Step 12.1.10.1

Multiply $- 1$ by $2$ .

$- \frac{2}{e} + 1 + 2 - 2 \frac{2}{e}$

Step 12.1.10.2

Combine $- 2$ and $\frac{2}{e}$ .

$- \frac{2}{e} + 1 + 2 + \frac{- 2 \cdot 2}{e}$

Step 12.1.10.3

Multiply $- 2$ by $2$ .

$- \frac{2}{e} + 1 + 2 + \frac{- 4}{e}$

Step 12.1.11

Move the negative in front of the fraction.

$- \frac{2}{e} + 1 + 2 - \frac{4}{e}$

Step 12.2

Combine the numerators over the common denominator.

$1 + 2 + \frac{- 2 - 4}{e}$

Step 12.3

Subtract $4$ from $- 2$ .

$1 + 2 + \frac{- 6}{e}$

Step 12.4

Move the negative in front of the fraction.

$1 + 2 - \frac{6}{e}$

Step 12.5

Add $1$ and $2$ .

$3 - \frac{6}{e}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$3 - \frac{6}{e}$

Decimal Form:

$0.79272335 \dots$

Step 14