Calculus Examples

$\int_{0}^{π} (\sin (x) - 6) d x$

Step 1

Remove parentheses.

$\int_{0}^{π} \sin (x) - 6 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify the answer.

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Step 5.1

Combine $- \cos (x)]_{0}^{π}$ and $- 6 x]_{0}^{π}$ .

$- \cos (x) - 6 x]_{0}^{π}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \cos (x) - 6 x$ at $π$ and at $0$ .

$(- \cos (π) - 6 π) - (- \cos (0) - 6 \cdot 0)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 6$ by $0$ .

$- \cos (π) - 6 π - (- \cos (0) + 0)$

Step 5.2.2.2

Add $- \cos (0)$ and $0$ .

$- \cos (π) - 6 π - - \cos (0)$

Step 5.2.2.3

Multiply $- 1$ by $- 1$ .

$- \cos (π) - 6 π + 1 \cos (0)$

Step 5.2.2.4

Multiply $\cos (0)$ by $1$ .

$- \cos (π) - 6 π + \cos (0)$

Step 5.3

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

$- \cos (π) - 6 π + 1$

Step 5.4

Simplify.

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Step 5.4.1

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.

$- - \cos (0) - 6 π + 1$

Step 5.4.2

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

$- (- 1 \cdot 1) - 6 π + 1$

Step 5.4.3

Multiply $- 1$ by $1$ .

$- - 1 - 6 π + 1$

Step 5.4.4

Multiply $- 1$ by $- 1$ .

$1 - 6 π + 1$

Step 5.4.5

Add $1$ and $1$ .

$2 - 6 π$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 - 6 π$

Decimal Form:

$- 16.84955592 \dots$