Calculus Examples

$\int_{0}^{π} 2 e^{x} + 6 \sin (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{π} 2 e^{x} d x + \int_{0}^{π} 6 \sin (x) d x$

Step 2

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

$2 \int_{0}^{π} e^{x} d x + \int_{0}^{π} 6 \sin (x) d x$

Step 3

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

$2 (e^{x}]_{0}^{π}) + \int_{0}^{π} 6 \sin (x) d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$2 (e^{x}]_{0}^{π}) + 6 \int_{0}^{π} \sin (x) d x$

Step 5

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

$2 (e^{x}]_{0}^{π}) + 6 (- \cos (x)]_{0}^{π})$

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

Evaluate $e^{x}$ at $π$ and at $0$ .

$2 ((e^{π}) - e^{0}) + 6 (- \cos (x)]_{0}^{π})$

Step 6.1.2

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

$2 (e^{π} - e^{0}) + 6 (- \cos (π) + \cos (0))$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Anything raised to $0$ is $1$ .

$2 (e^{π} - 1 \cdot 1) + 6 (- \cos (π) + \cos (0))$

Step 6.1.3.2

Multiply $- 1$ by $1$ .

$2 (e^{π} - 1) + 6 (- \cos (π) + \cos (0))$

Step 6.2

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

$2 (e^{π} - 1) + 6 (- \cos (π) + 1)$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the distributive property.

$2 e^{π} + 2 \cdot - 1 + 6 (- \cos (π) + 1)$

Step 7.1.2

Multiply $2$ by $- 1$ .

$2 e^{π} - 2 + 6 (- \cos (π) + 1)$

Step 7.1.3

Simplify each term.

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Step 7.1.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 e^{π} - 2 + 6 (- - \cos (0) + 1)$

Step 7.1.3.2

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

$2 e^{π} - 2 + 6 (- (- 1 \cdot 1) + 1)$

Step 7.1.3.3

Multiply $- (- 1 \cdot 1)$ .

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Step 7.1.3.3.1

Multiply $- 1$ by $1$ .

$2 e^{π} - 2 + 6 (- - 1 + 1)$

Step 7.1.3.3.2

Multiply $- 1$ by $- 1$ .

$2 e^{π} - 2 + 6 (1 + 1)$

Step 7.1.4

Add $1$ and $1$ .

$2 e^{π} - 2 + 6 \cdot 2$

Step 7.1.5

Multiply $6$ by $2$ .

$2 e^{π} - 2 + 12$

Step 7.2

Add $- 2$ and $12$ .

$2 e^{π} + 10$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 e^{π} + 10$

Decimal Form:

$56.28138526 \dots$