Calculus Examples

$\int_{0}^{π} (2 + \cos (x)) d x$

Step 1

Remove parentheses.

$\int_{0}^{π} 2 + \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

$2 x]_{0}^{π} + \sin (x)]_{0}^{π}$

Step 5

Simplify the answer.

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Step 5.1

Combine $2 x]_{0}^{π}$ and $\sin (x)]_{0}^{π}$ .

$2 x + \sin (x)]_{0}^{π}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $2$ by $0$ .

$2 π + \sin (π) - (0 + \sin (0))$

Step 5.2.2.2

Add $0$ and $\sin (0)$ .

$2 π + \sin (π) - \sin (0)$

Step 5.3

Simplify.

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Step 5.3.1

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

$2 π + \sin (π) - 0$

Step 5.3.2

Multiply $- 1$ by $0$ .

$2 π + \sin (π) + 0$

Step 5.3.3

Add $2 π + \sin (π)$ and $0$ .

$2 π + \sin (π)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$2 π + \sin (0)$

Step 6.1.2

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

$2 π + 0$

Step 6.2

Add $2 π$ and $0$ .

$2 π$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 π$

Decimal Form:

$6.28318530 \dots$