Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Simplify.

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Step 6.1.3.1

Factor $- 1$ out of $- \frac{π}{2}$ .

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

Step 6.1.3.2

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

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

Step 6.1.3.3

Raise $- 1$ to the power of $2$ .

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

Step 6.1.3.4

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

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

Step 6.1.3.5

Combine the numerators over the common denominator.

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

Step 6.1.3.6

Subtract ${(\frac{π}{2})}^{2}$ from ${(\frac{π}{2})}^{2}$ .

$2 (\frac{0}{2}) + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.7

Cancel the common factor of $0$ and $2$ .

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Step 6.1.3.7.1

Factor $2$ out of $0$ .

$2 \frac{2 (0)}{2} + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.7.2

Cancel the common factors.

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Step 6.1.3.7.2.1

Factor $2$ out of $2$ .

$2 \frac{2 \cdot 0}{2 \cdot 1} + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.7.2.2

Cancel the common factor.

$2 \frac{2 \cdot 0}{2 \cdot 1} + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.7.2.3

Rewrite the expression.

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

Step 6.1.3.7.2.4

Divide $0$ by $1$ .

$2 \cdot 0 + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.8

Multiply $2$ by $0$ .

$0 + \sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.1.3.9

Add $0$ and $\sin (\frac{π}{2}) - \sin (- \frac{π}{2})$ .

$\sin (\frac{π}{2}) - \sin (- \frac{π}{2})$

Step 6.2

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

$1 - \sin (- \frac{π}{2})$

Step 6.3

Simplify.

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Step 6.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$1 - \sin (\frac{3 π}{2})$

Step 6.3.2

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

$1 - - \sin (\frac{π}{2})$

Step 6.3.3

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

$1 - (- 1 \cdot 1)$

Step 6.3.4

Multiply $- 1$ by $1$ .

$1 - - 1$

Step 6.3.5

Multiply $- 1$ by $- 1$ .

$1 + 1$

Step 6.3.6

Add $1$ and $1$ .

$2$