Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify.

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

Step 5.2.3.2

Anything raised to $0$ is $1$ .

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

Step 5.2.3.3

Multiply $- 1$ by $1$ .

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

Step 5.2.3.4

One to any power is one.

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

Step 5.2.3.5

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

$e - 1 - (\frac{1}{2} - \frac{0}{2})$

Step 5.2.3.6

Cancel the common factor of $0$ and $2$ .

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Step 5.2.3.6.1

Factor $2$ out of $0$ .

$e - 1 - (\frac{1}{2} - \frac{2 (0)}{2})$

Step 5.2.3.6.2

Cancel the common factors.

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Step 5.2.3.6.2.1

Factor $2$ out of $2$ .

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

Step 5.2.3.6.2.2

Cancel the common factor.

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

Step 5.2.3.6.2.3

Rewrite the expression.

$e - 1 - (\frac{1}{2} - \frac{0}{1})$

Step 5.2.3.6.2.4

Divide $0$ by $1$ .

$e - 1 - (\frac{1}{2} - 0)$

Step 5.2.3.7

Multiply $- 1$ by $0$ .

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

Step 5.2.3.8

Add $\frac{1}{2}$ and $0$ .

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

Step 5.2.3.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.3.10

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

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

Step 5.2.3.11

Combine the numerators over the common denominator.

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

Step 5.2.3.12

Simplify the numerator.

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Step 5.2.3.12.1

Multiply $- 1$ by $2$ .

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

Step 5.2.3.12.2

Subtract $1$ from $- 2$ .

$e + \frac{- 3}{2}$

Step 5.2.3.13

Move the negative in front of the fraction.

$e - \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$e - \frac{3}{2}$

Decimal Form:

$1.21828182 \dots$

Step 7