Calculus Examples

$\int_{- 0.5}^{0.5} 4 \cos (π x) - (12 x^{2} - 3) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 0.5}^{0.5} 4 \cos (π x) d x + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{- 0.5}^{0.5} \cos (π x) d x + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 3

Let $u = π x$ . Then $d u = π d x$ , so $\frac{1}{π} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = π x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 3.1.2

Since $π$ is constant with respect to $x$ , the derivative of $π x$ with respect to $x$ is $π \frac{d}{d x} [x]$ .

$π \frac{d}{d x} [x]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$π \cdot 1$

Step 3.1.4

Multiply $π$ by $1$ .

$π$

Step 3.2

Substitute the lower limit in for $x$ in $u = π x$ .

$u_{lower} = π \cdot - 0.5$

Step 3.3

Multiply $π$ by $- 0.5$ .

$u_{lower} = - 1.57079632$

Step 3.4

Substitute the upper limit in for $x$ in $u = π x$ .

$u_{upper} = π \cdot 0.5$

Step 3.5

Multiply $π$ by $0.5$ .

$u_{upper} = 1.57079632$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 1.57079632$

$u_{upper} = 1.57079632$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$4 \int_{- 1.57079632}^{1.57079632} \cos (u) \frac{1}{π} d u + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 4

Combine $\cos (u)$ and $\frac{1}{π}$ .

$4 \int_{- 1.57079632}^{1.57079632} \frac{\cos (u)}{π} d u + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 5

Since $\frac{1}{π}$ is constant with respect to $u$ , move $\frac{1}{π}$ out of the integral.

$4 (\frac{1}{π} \int_{- 1.57079632}^{1.57079632} \cos (u) d u) + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 6

Combine $\frac{1}{π}$ and $4$ .

$\frac{4}{π} \int_{- 1.57079632}^{1.57079632} \cos (u) d u + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 7

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

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} + \int_{- 0.5}^{0.5} - (12 x^{2} - 3) d x$

Step 8

Multiply $- (12 x^{2} - 3)$ .

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} + \int_{- 0.5}^{0.5} - (12 x^{2}) - - 3 d x$

Step 9

Simplify.

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Step 9.1

Multiply $12$ by $- 1$ .

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} + \int_{- 0.5}^{0.5} - 12 x^{2} - - 3 d x$

Step 9.2

Multiply $- 1$ by $- 3$ .

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} + \int_{- 0.5}^{0.5} - 12 x^{2} + 3 d x$

Step 10

Split the single integral into multiple integrals.

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} + \int_{- 0.5}^{0.5} - 12 x^{2} d x + \int_{- 0.5}^{0.5} 3 d x$

Step 11

Since $- 12$ is constant with respect to $x$ , move $- 12$ out of the integral.

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} - 12 \int_{- 0.5}^{0.5} x^{2} d x + \int_{- 0.5}^{0.5} 3 d x$

Step 12

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} - 12 (\frac{1}{3} x^{3}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 3 d x$

Step 13

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} - 12 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 3 d x$

Step 14

Apply the constant rule.

$\frac{4}{π} \sin (u)]_{- 1.57079632}^{1.57079632} - 12 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 3 x]_{- 0.5}^{0.5}$

Step 15

Substitute and simplify.

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Step 15.1

Evaluate $\sin (u)$ at $1.57079632$ and at $- 1.57079632$ .

$\frac{4}{π} ((\sin (1.57079632)) - \sin (- 1.57079632)) - 12 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 3 x]_{- 0.5}^{0.5}$

Step 15.2

Evaluate $\frac{x^{3}}{3}$ at $0.5$ and at $- 0.5$ .

$\frac{4}{π} ((\sin (1.57079632)) - \sin (- 1.57079632)) - 12 ((\frac{{0.5}^{3}}{3}) - \frac{{(- 0.5)}^{3}}{3}) + 3 x]_{- 0.5}^{0.5}$

Step 15.3

Evaluate $3 x$ at $0.5$ and at $- 0.5$ .

$\frac{4}{π} ((\sin (1.57079632)) - \sin (- 1.57079632)) - 12 (\frac{{0.5}^{3}}{3} - \frac{{(- 0.5)}^{3}}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4

Simplify.

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Step 15.4.1

Raise $0.5$ to the power of $3$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.125}{3} - \frac{{(- 0.5)}^{3}}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.2

Raise $- 0.5$ to the power of $3$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.125}{3} - \frac{- 0.125}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.3

Move the negative in front of the fraction.

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.125}{3} - - \frac{0.125}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.4

Multiply $- 1$ by $- 1$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.125}{3} + 1 (\frac{0.125}{3})) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.5

Multiply $\frac{0.125}{3}$ by $1$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.125}{3} + \frac{0.125}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.6

Combine the numerators over the common denominator.

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 \frac{0.125 + 0.125}{3} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.7

Add $0.125$ and $0.125$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 12 (\frac{0.25}{3}) + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.8

Combine $- 12$ and $\frac{0.25}{3}$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{- 12 \cdot 0.25}{3} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.9

Multiply $- 12$ by $0.25$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{- 3}{3} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.10

Cancel the common factor of $- 3$ and $3$ .

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Step 15.4.10.1

Factor $3$ out of $- 3$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{3 \cdot - 1}{3} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.10.2

Cancel the common factors.

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Step 15.4.10.2.1

Factor $3$ out of $3$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{3 \cdot - 1}{3 (1)} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.10.2.2

Cancel the common factor.

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{3 \cdot - 1}{3 \cdot 1} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.10.2.3

Rewrite the expression.

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + \frac{- 1}{1} + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.10.2.4

Divide $- 1$ by $1$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 1 + 3 \cdot 0.5 - 3 \cdot - 0.5$

Step 15.4.11

Multiply $3$ by $0.5$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 1 + 1.5 - 3 \cdot - 0.5$

Step 15.4.12

Multiply $- 3$ by $- 0.5$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 1 + 1.5 + 1.5$

Step 15.4.13

Add $1.5$ and $1.5$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) - 1 + 3$

Step 15.4.14

Add $- 1$ and $3$ .

$\frac{4}{π} (\sin (1.57079632) - \sin (- 1.57079632)) + 2$

Step 16

Simplify.

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Step 16.1

Simplify each term.

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Step 16.1.1

Apply the distributive property.

$\frac{4}{π} \sin (1.57079632) + \frac{4}{π} (- \sin (- 1.57079632)) + 2$

Step 16.1.2

Combine $\frac{4}{π}$ and $\sin (1.57079632)$ .

$\frac{4 \sin (1.57079632)}{π} + \frac{4}{π} (- \sin (- 1.57079632)) + 2$

Step 16.1.3

Combine $\frac{4}{π}$ and $\sin (- 1.57079632)$ .

$\frac{4 \sin (1.57079632)}{π} - \frac{4 \sin (- 1.57079632)}{π} + 2$

Step 16.1.4

Simplify each term.

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Step 16.1.4.1

Evaluate $\sin (1.57079632)$ .

$\frac{4 \cdot 1}{π} - \frac{4 \sin (- 1.57079632)}{π} + 2$

Step 16.1.4.2

Multiply $4$ by $1$ .

$\frac{4}{π} - \frac{4 \sin (- 1.57079632)}{π} + 2$

Step 16.1.4.3

Evaluate $\sin (- 1.57079632)$ .

$\frac{4}{π} - \frac{4 \cdot - 1}{π} + 2$

Step 16.1.4.4

Multiply $4$ by $- 1$ .

$\frac{4}{π} - \frac{- 4}{π} + 2$

Step 16.1.4.5

Move the negative in front of the fraction.

$\frac{4}{π} - - \frac{4}{π} + 2$

Step 16.1.4.6

Multiply $- - \frac{4}{π}$ .

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Step 16.1.4.6.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{π} + 1 \frac{4}{π} + 2$

Step 16.1.4.6.2

Multiply $\frac{4}{π}$ by $1$ .

$\frac{4}{π} + \frac{4}{π} + 2$

Step 16.1.5

Combine the numerators over the common denominator.

$\frac{4 + 4}{π} + 2$

Step 16.1.6

Add $4$ and $4$ .

$\frac{8}{π} + 2$

Step 16.2

Add $\frac{8}{π}$ and $2$ .

$4.54647908$