Calculus Examples

$\int_{1}^{3} 2 e^{x} - 1 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} 2 e^{x} d x + \int_{1}^{3} - 1 d x$

Step 2

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

$2 \int_{1}^{3} e^{x} d x + \int_{1}^{3} - 1 d x$

Step 3

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

$2 (e^{x}]_{1}^{3}) + \int_{1}^{3} - 1 d x$

Step 4

Apply the constant rule.

$2 (e^{x}]_{1}^{3}) + - x]_{1}^{3}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $e^{x}$ at $3$ and at $1$ .

$2 ((e^{3}) - e^{1}) + - x]_{1}^{3}$

Step 5.2

Evaluate $- x$ at $3$ and at $1$ .

$2 (e^{3} - e^{1}) + - 1 \cdot 3 + 1 \cdot 1$

Step 5.3

Simplify.

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Step 5.3.1

Simplify.

$2 (e^{3} - e) - 1 \cdot 3 + 1 \cdot 1$

Step 5.3.2

Multiply $- 1$ by $3$ .

$2 (e^{3} - e) - 3 + 1 \cdot 1$

Step 5.3.3

Multiply $1$ by $1$ .

$2 (e^{3} - e) - 3 + 1$

Step 5.3.4

Add $- 3$ and $1$ .

$2 (e^{3} - e) - 2$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$2 e^{3} + 2 (- e) - 2$

Step 6.2

Multiply $- 1$ by $2$ .

$2 e^{3} - 2 e - 2$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 e^{3} - 2 e - 2$

Decimal Form:

$32.73451018 \dots$

Step 8