Calculus Examples

$\int_{\frac{π}{2}}^{π} x - \sin (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{\frac{π}{2}}^{π} x d x + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 2

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2}]_{\frac{π}{2}}^{π} + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} x^{2}]_{\frac{π}{2}}^{π} - \int_{\frac{π}{2}}^{π} \sin (x) d x$

Step 4

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

$\frac{1}{2} x^{2}]_{\frac{π}{2}}^{π} - (- \cos (x)]_{\frac{π}{2}}^{π})$

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $\frac{1}{2} x^{2}$ at $π$ and at $\frac{π}{2}$ .

$(\frac{1}{2} π^{2}) - \frac{1}{2} {(\frac{π}{2})}^{2} - (- \cos (x)]_{\frac{π}{2}}^{π})$

Step 5.1.2

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

$\frac{1}{2} π^{2} - \frac{1}{2} {(\frac{π}{2})}^{2} - (- \cos (π) + \cos (\frac{π}{2}))$

Step 5.1.3

Combine $\frac{1}{2}$ and $π^{2}$ .

$\frac{π^{2}}{2} - \frac{1}{2} {(\frac{π}{2})}^{2} - (- \cos (π) + \cos (\frac{π}{2}))$

Step 5.2

Simplify.

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Step 5.2.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

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

Step 5.2.2

Add $- \cos (π)$ and $0$ .

$\frac{π^{2}}{2} - \frac{1}{2} {(\frac{π}{2})}^{2} - - \cos (π)$

Step 5.2.3

Multiply $- 1$ by $- 1$ .

$\frac{π^{2}}{2} - \frac{1}{2} {(\frac{π}{2})}^{2} + 1 \cos (π)$

Step 5.2.4

Multiply $\cos (π)$ by $1$ .

$\frac{π^{2}}{2} - \frac{1}{2} {(\frac{π}{2})}^{2} + \cos (π)$

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{π}{2}$ .

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

Step 5.3.2

Raise $2$ to the power of $2$ .

$\frac{π^{2}}{2} - \frac{1}{2} \cdot \frac{π^{2}}{4} + \cos (π)$

Step 5.3.3

Multiply $- \frac{1}{2} \cdot \frac{π^{2}}{4}$ .

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Step 5.3.3.1

Multiply $\frac{π^{2}}{4}$ by $\frac{1}{2}$ .

$\frac{π^{2}}{2} - \frac{π^{2}}{4 \cdot 2} + \cos (π)$

Step 5.3.3.2

Multiply $4$ by $2$ .

$\frac{π^{2}}{2} - \frac{π^{2}}{8} + \cos (π)$

Step 5.3.4

To write $\frac{π^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π^{2}}{2} \cdot \frac{4}{4} - \frac{π^{2}}{8} + \cos (π)$

Step 5.3.5

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.5.1

Multiply $\frac{π^{2}}{2}$ by $\frac{4}{4}$ .

$\frac{π^{2} \cdot 4}{2 \cdot 4} - \frac{π^{2}}{8} + \cos (π)$

Step 5.3.5.2

Multiply $2$ by $4$ .

$\frac{π^{2} \cdot 4}{8} - \frac{π^{2}}{8} + \cos (π)$

Step 5.3.6

Combine the numerators over the common denominator.

$\frac{π^{2} \cdot 4 - π^{2}}{8} + \cos (π)$

Step 5.3.7

Subtract $π^{2}$ from $π^{2} \cdot 4$ .

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Step 5.3.7.1

Reorder $π^{2}$ and $4$ .

$\frac{4 \cdot π^{2} - π^{2}}{8} + \cos (π)$

Step 5.3.7.2

Subtract $π^{2}$ from $4 \cdot π^{2}$ .

$\frac{3 \cdot π^{2}}{8} + \cos (π)$

$\frac{3 π^{2}}{8} + \cos (π)$

Step 6

Simplify each term.

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Step 6.1

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.

$\frac{3 π^{2}}{8} - \cos (0)$

Step 6.2

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

$\frac{3 π^{2}}{8} - 1 \cdot 1$

Step 6.3

Multiply $- 1$ by $1$ .

$\frac{3 π^{2}}{8} - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 π^{2}}{8} - 1$

Decimal Form:

$2.70110165 \dots$