Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} e d x + \int_{1}^{2} - x d x$

Step 2

Apply the constant rule.

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $e x$ at $2$ and at $1$ .

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

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Move $2$ to the left of $e$ .

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

Step 5.2.3.2

Multiply $- 1$ by $1$ .

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

Step 5.2.3.3

Subtract $e$ from $2 e$ .

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

Step 5.2.3.4

Raise $2$ to the power of $2$ .

$e - (\frac{4}{2} - \frac{1^{2}}{2})$

Step 5.2.3.5

Cancel the common factor of $4$ and $2$ .

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Step 5.2.3.5.1

Factor $2$ out of $4$ .

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

Step 5.2.3.5.2

Cancel the common factors.

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Step 5.2.3.5.2.1

Factor $2$ out of $2$ .

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

Step 5.2.3.5.2.2

Cancel the common factor.

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

Step 5.2.3.5.2.3

Rewrite the expression.

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

Step 5.2.3.5.2.4

Divide $2$ by $1$ .

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

Step 5.2.3.6

One to any power is one.

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

Step 5.2.3.7

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

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

Step 5.2.3.8

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

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

Step 5.2.3.9

Combine the numerators over the common denominator.

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

Step 5.2.3.10

Simplify the numerator.

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Step 5.2.3.10.1

Multiply $2$ by $2$ .

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

Step 5.2.3.10.2

Subtract $1$ from $4$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.21828182 \dots$

Step 7