Calculus Examples

$\int_{0}^{3} (2 \sin (x) - e^{x}) d x$

Step 1

Remove parentheses.

$\int_{0}^{3} 2 \sin (x) - e^{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{3} 2 \sin (x) d x + \int_{0}^{3} - e^{x} d x$

Step 3

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

$2 \int_{0}^{3} \sin (x) d x + \int_{0}^{3} - e^{x} d x$

Step 4

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$2 (- \cos (x)]_{0}^{3}) + \int_{0}^{3} - e^{x} d x$

Step 5

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

$2 (- \cos (x)]_{0}^{3}) - \int_{0}^{3} e^{x} d x$

Step 6

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

$2 (- \cos (x)]_{0}^{3}) - (e^{x}]_{0}^{3})$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

Evaluate $- \cos (x)$ at $3$ and at $0$ .

$2 ((- \cos (3)) + \cos (0)) - (e^{x}]_{0}^{3})$

Step 7.1.2

Evaluate $e^{x}$ at $3$ and at $0$ .

$2 (- \cos (3) + \cos (0)) - (e^{3} - e^{0})$

Step 7.1.3

Simplify.

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Step 7.1.3.1

Anything raised to $0$ is $1$ .

$2 (- \cos (3) + \cos (0)) - (e^{3} - 1 \cdot 1)$

Step 7.1.3.2

Multiply $- 1$ by $1$ .

$2 (- \cos (3) + \cos (0)) - (e^{3} - 1)$

Step 7.2

The exact value of $\cos (0)$ is $1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Evaluate $\cos (3)$ .

$2 (- - 0.98999249 + 1) - (e^{3} - 1)$

Step 7.3.2

Multiply $- 1$ by $- 0.98999249$ .

$2 (0.98999249 + 1) - (e^{3} - 1)$

Step 7.3.3

Add $0.98999249$ and $1$ .

$2 \cdot 1.98999249 - (e^{3} - 1)$

Step 7.3.4

Multiply $2$ by $1.98999249$ .

$3.97998499 - (e^{3} - 1)$

Step 7.3.5

Apply the distributive property.

$3.97998499 - e^{3} - - 1$

Step 7.3.6

Multiply $- 1$ by $- 1$ .

$3.97998499 - e^{3} + 1$

Step 7.3.7

Subtract $e^{3}$ from $3.97998499$ .

$- 16.10555192 + 1$

Step 7.3.8

Add $- 16.10555192$ and $1$ .

$- 15.10555192$