Calculus Examples

$\int_{0}^{2} \sin (π x) - (x^{3} - 4 x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{2} \sin (π x) d x + \int_{0}^{2} - (x^{3} - 4 x) d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $π$ by $1$ .

$π$

Step 2.2

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

$u_{lower} = π \cdot 0$

Step 2.3

Multiply $π$ by $0$ .

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = π \cdot 2$

Step 2.5

Move $2$ to the left of $π$ .

$u_{upper} = 2 π$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 2.7

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

$\int_{0}^{2 π} \sin (u) \frac{1}{π} d u + \int_{0}^{2} - (x^{3} - 4 x) d x$

Step 3

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

$\int_{0}^{2 π} \frac{\sin (u)}{π} d u + \int_{0}^{2} - (x^{3} - 4 x) d x$

Step 4

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

$\frac{1}{π} \int_{0}^{2 π} \sin (u) d u + \int_{0}^{2} - (x^{3} - 4 x) d x$

Step 5

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

$\frac{1}{π} - \cos (u)]_{0}^{2 π} + \int_{0}^{2} - (x^{3} - 4 x) d x$

Step 6

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

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - \int_{0}^{2} x^{3} - 4 x d x$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\int_{0}^{2} x^{3} d x + \int_{0}^{2} - 4 x d x)$

Step 8

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\frac{1}{4} x^{4}]_{0}^{2} + \int_{0}^{2} - 4 x d x)$

Step 9

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\frac{x^{4}}{4}]_{0}^{2} + \int_{0}^{2} - 4 x d x)$

Step 10

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

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\frac{x^{4}}{4}]_{0}^{2} - 4 \int_{0}^{2} x d x)$

Step 11

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

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\frac{x^{4}}{4}]_{0}^{2} - 4 (\frac{1}{2} x^{2}]_{0}^{2}))$

Step 12

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

$\frac{1}{π} - \cos (u)]_{0}^{2 π} - (\frac{x^{4}}{4}]_{0}^{2} - 4 (\frac{x^{2}}{2}]_{0}^{2}))$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $- \cos (u)$ at $2 π$ and at $0$ .

$\frac{1}{π} ((- \cos (2 π)) + \cos (0)) - (\frac{x^{4}}{4}]_{0}^{2} - 4 (\frac{x^{2}}{2}]_{0}^{2}))$

Step 13.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $0$ .

$\frac{1}{π} ((- \cos (2 π)) + \cos (0)) - ((\frac{2^{4}}{4}) - \frac{0^{4}}{4} - 4 (\frac{x^{2}}{2}]_{0}^{2}))$

Step 13.3

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $0$ .

$\frac{1}{π} ((- \cos (2 π)) + \cos (0)) - (\frac{2^{4}}{4} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4

Simplify.

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Step 13.4.1

Raise $2$ to the power of $4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (\frac{16}{4} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.2

Cancel the common factor of $16$ and $4$ .

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Step 13.4.2.1

Factor $4$ out of $16$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (\frac{4 \cdot 4}{4} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $4$ out of $4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (\frac{4 \cdot 4}{4 (1)} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.2.2.2

Cancel the common factor.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (\frac{4 \cdot 4}{4 \cdot 1} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.2.2.3

Rewrite the expression.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (\frac{4}{1} - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.2.2.4

Divide $4$ by $1$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{0^{4}}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{0}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.4

Cancel the common factor of $0$ and $4$ .

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Step 13.4.4.1

Factor $4$ out of $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{4 (0)}{4} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.4.2

Cancel the common factors.

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Step 13.4.4.2.1

Factor $4$ out of $4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{4 \cdot 0}{4 \cdot 1} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.4.2.2

Cancel the common factor.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{4 \cdot 0}{4 \cdot 1} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.4.2.3

Rewrite the expression.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - \frac{0}{1} - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.4.2.4

Divide $0$ by $1$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 0 - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.5

Multiply $- 1$ by $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 + 0 - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.6

Add $4$ and $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}))$

Step 13.4.7

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

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{4}{2} - \frac{0^{2}}{2}))$

Step 13.4.8

Cancel the common factor of $4$ and $2$ .

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Step 13.4.8.1

Factor $2$ out of $4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}))$

Step 13.4.8.2

Cancel the common factors.

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Step 13.4.8.2.1

Factor $2$ out of $2$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}))$

Step 13.4.8.2.2

Cancel the common factor.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}))$

Step 13.4.8.2.3

Rewrite the expression.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (\frac{2}{1} - \frac{0^{2}}{2}))$

Step 13.4.8.2.4

Divide $2$ by $1$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{0^{2}}{2}))$

Step 13.4.9

Raising $0$ to any positive power yields $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{0}{2}))$

Step 13.4.10

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

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Step 13.4.10.1

Factor $2$ out of $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{2 (0)}{2}))$

Step 13.4.10.2

Cancel the common factors.

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Step 13.4.10.2.1

Factor $2$ out of $2$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 13.4.10.2.2

Cancel the common factor.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 13.4.10.2.3

Rewrite the expression.

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - \frac{0}{1}))$

Step 13.4.10.2.4

Divide $0$ by $1$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 - 0))$

Step 13.4.11

Multiply $- 1$ by $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 (2 + 0))$

Step 13.4.12

Add $2$ and $0$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 4 \cdot 2)$

Step 13.4.13

Multiply $- 4$ by $2$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - (4 - 8)$

Step 13.4.14

Subtract $8$ from $4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) - - 4$

Step 13.4.15

Multiply $- 1$ by $- 4$ .

$\frac{1}{π} (- \cos (2 π) + \cos (0)) + 4$

Step 14

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

$\frac{1}{π} (- \cos (2 π) + 1) + 4$

Step 15

Simplify.

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Step 15.1

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

$\frac{1}{π} (- \cos (0) + 1) + 4$

Step 15.2

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

$\frac{1}{π} (- 1 \cdot 1 + 1) + 4$

Step 15.3

Multiply $- 1$ by $1$ .

$\frac{1}{π} (- 1 + 1) + 4$

Step 15.4

Add $- 1$ and $1$ .

$\frac{1}{π} \cdot 0 + 4$

Step 15.5

Multiply $\frac{1}{π}$ by $0$ .

$0 + 4$

Step 15.6

Add $0$ and $4$ .

$4$