Calculus Examples

$\int_{\frac{π}{2}}^{π} 5 \cos (x) d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{\frac{π}{2}}^{π} \cos (x) d x$

Step 2

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

$5 (\sin (x)]_{\frac{π}{2}}^{π})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\sin (x)$ at $π$ and at $\frac{π}{2}$ .

$5 (\sin (π) - \sin (\frac{π}{2}))$

Step 3.2

Simplify.

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Step 3.2.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$5 (\sin (π) - 1 \cdot 1)$

Step 3.2.2

Multiply $- 1$ by $1$ .

$5 (\sin (π) - 1)$

Step 3.3

Simplify.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

$5 (\sin (0) - 1)$

Step 3.3.1.2

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

$5 (0 - 1)$

Step 3.3.2

Subtract $1$ from $0$ .

$5 \cdot - 1$

Step 3.3.3

Multiply $5$ by $- 1$ .

$- 5$