Calculus Examples

$\int_{0}^{2 π} 2 - \cos (x) - \cos (x) d x$

Step 1

Subtract $\cos (x)$ from $- \cos (x)$ .

$\int_{0}^{2 π} 2 - 2 \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2 π} 2 d x + \int_{0}^{2 π} - 2 \cos (x) d x$

Step 3

Apply the constant rule.

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

Step 4

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

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

Step 5

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

$2 x]_{0}^{2 π} - 2 (\sin (x)]_{0}^{2 π})$

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

Evaluate $2 x$ at $2 π$ and at $0$ .

$(2 (2 π)) - 2 \cdot 0 - 2 (\sin (x)]_{0}^{2 π})$

Step 6.1.2

Evaluate $\sin (x)$ at $2 π$ and at $0$ .

$2 (2 π) - 2 \cdot 0 - 2 (\sin (2 π) - \sin (0))$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Multiply $2$ by $2$ .

$4 π - 2 \cdot 0 - 2 (\sin (2 π) - \sin (0))$

Step 6.1.3.2

Multiply $- 2$ by $0$ .

$4 π + 0 - 2 (\sin (2 π) - \sin (0))$

Step 6.1.3.3

Add $4 π$ and $0$ .

$4 π - 2 (\sin (2 π) - \sin (0))$

Step 6.2

Simplify.

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Step 6.2.1

The exact value of $\sin (0)$ is $0$ .

$4 π - 2 (\sin (2 π) - 0)$

Step 6.2.2

Multiply $- 1$ by $0$ .

$4 π - 2 (\sin (2 π) + 0)$

Step 6.2.3

Add $\sin (2 π)$ and $0$ .

$4 π - 2 \sin (2 π)$

Step 6.3

Simplify.

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Step 6.3.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$4 π - 2 \sin (0)$

Step 6.3.2

The exact value of $\sin (0)$ is $0$ .

$4 π - 2 \cdot 0$

Step 6.3.3

Multiply $- 2$ by $0$ .

$4 π + 0$

Step 6.3.4

Add $4 π$ and $0$ .

$4 π$

Step 7

The result can be shown in multiple forms.

Exact Form:

$4 π$

Decimal Form:

$12.56637061 \dots$