Calculus Examples

$\int_{0}^{9} 0.5 x + \sin (2 x) + 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{9} 0.5 x d x + \int_{0}^{9} \sin (2 x) d x + \int_{0}^{9} 3 d x$

Step 2

Since $0.5$ is constant with respect to $x$ , move $0.5$ out of the integral.

$0.5 \int_{0}^{9} x d x + \int_{0}^{9} \sin (2 x) d x + \int_{0}^{9} 3 d x$

Step 3

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

$0.5 (\frac{1}{2} x^{2}]_{0}^{9}) + \int_{0}^{9} \sin (2 x) d x + \int_{0}^{9} 3 d x$

Step 4

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

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \int_{0}^{9} \sin (2 x) d x + \int_{0}^{9} 3 d x$

Step 5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 5.1.3

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

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 9$

Step 5.5

Multiply $2$ by $9$ .

$u_{upper} = 18$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = 18$

Step 5.7

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

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \int_{0}^{18} \sin (u) \frac{1}{2} d u + \int_{0}^{9} 3 d x$

Step 6

Combine $\sin (u)$ and $\frac{1}{2}$ .

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \int_{0}^{18} \frac{\sin (u)}{2} d u + \int_{0}^{9} 3 d x$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \frac{1}{2} \int_{0}^{18} \sin (u) d u + \int_{0}^{9} 3 d x$

Step 8

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

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \frac{1}{2} - \cos (u)]_{0}^{18} + \int_{0}^{9} 3 d x$

Step 9

Apply the constant rule.

$0.5 (\frac{x^{2}}{2}]_{0}^{9}) + \frac{1}{2} - \cos (u)]_{0}^{18} + 3 x]_{0}^{9}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{x^{2}}{2}$ at $9$ and at $0$ .

$0.5 ((\frac{9^{2}}{2}) - \frac{0^{2}}{2}) + \frac{1}{2} - \cos (u)]_{0}^{18} + 3 x]_{0}^{9}$

Step 10.2

Evaluate $- \cos (u)$ at $18$ and at $0$ .

$0.5 ((\frac{9^{2}}{2}) - \frac{0^{2}}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 x]_{0}^{9}$

Step 10.3

Evaluate $3 x$ at $9$ and at $0$ .

$0.5 (\frac{9^{2}}{2} - \frac{0^{2}}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4

Simplify.

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Step 10.4.1

Raise $9$ to the power of $2$ .

$0.5 (\frac{81}{2} - \frac{0^{2}}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.2

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

$0.5 (\frac{81}{2} - \frac{0}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.3

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

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Step 10.4.3.1

Factor $2$ out of $0$ .

$0.5 (\frac{81}{2} - \frac{2 (0)}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.3.2

Cancel the common factors.

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Step 10.4.3.2.1

Factor $2$ out of $2$ .

$0.5 (\frac{81}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.3.2.2

Cancel the common factor.

$0.5 (\frac{81}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.3.2.3

Rewrite the expression.

$0.5 (\frac{81}{2} - \frac{0}{1}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.3.2.4

Divide $0$ by $1$ .

$0.5 (\frac{81}{2} - 0) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.4

Multiply $- 1$ by $0$ .

$0.5 (\frac{81}{2} + 0) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.5

Add $\frac{81}{2}$ and $0$ .

$0.5 (\frac{81}{2}) + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.6

Combine $0.5$ and $\frac{81}{2}$ .

$\frac{0.5 \cdot 81}{2} + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.7

Multiply $0.5$ by $81$ .

$\frac{40.5}{2} + \frac{1}{2} ((- \cos (18)) + \cos (0)) + 3 \cdot 9 - 3 \cdot 0$

Step 10.4.8

Multiply $3$ by $9$ .

$\frac{40.5}{2} + \frac{1}{2} (- \cos (18) + \cos (0)) + 27 - 3 \cdot 0$

Step 10.4.9

Multiply $- 3$ by $0$ .

$\frac{40.5}{2} + \frac{1}{2} (- \cos (18) + \cos (0)) + 27 + 0$

Step 10.4.10

Add $27$ and $0$ .

$\frac{40.5}{2} + \frac{1}{2} (- \cos (18) + \cos (0)) + 27$

Step 10.4.11

To write $27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (- \cos (18) + \cos (0)) + \frac{40.5}{2} + 27 \cdot \frac{2}{2}$

Step 10.4.12

Combine $27$ and $\frac{2}{2}$ .

$\frac{1}{2} (- \cos (18) + \cos (0)) + \frac{40.5}{2} + \frac{27 \cdot 2}{2}$

Step 10.4.13

Combine the numerators over the common denominator.

$\frac{1}{2} (- \cos (18) + \cos (0)) + \frac{40.5 + 27 \cdot 2}{2}$

Step 10.4.14

Multiply $27$ by $2$ .

$\frac{1}{2} (- \cos (18) + \cos (0)) + \frac{40.5 + 54}{2}$

Step 10.4.15

Add $40.5$ and $54$ .

$\frac{1}{2} (- \cos (18) + \cos (0)) + \frac{94.5}{2}$

Step 11

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

$\frac{1}{2} (- \cos (18) + 1) + \frac{94.5}{2}$

Step 12

Simplify.

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Step 12.1

Evaluate $\cos (18)$ .

$\frac{1}{2} (- 1 \cdot 0.6603167 + 1) + \frac{94.5}{2}$

Step 12.2

Multiply $- 1$ by $0.6603167$ .

$\frac{1}{2} (- 0.6603167 + 1) + \frac{94.5}{2}$

Step 12.3

Add $- 0.6603167$ and $1$ .

$\frac{1}{2} \cdot 0.33968329 + \frac{94.5}{2}$

Step 12.4

Combine $\frac{1}{2}$ and $0.33968329$ .

$\frac{0.33968329}{2} + \frac{94.5}{2}$

Step 12.5

Divide $0.33968329$ by $2$ .

$0.16984164 + \frac{94.5}{2}$

Step 12.6

Divide $94.5$ by $2$ .

$0.16984164 + 47.25$

Step 12.7

Add $0.16984164$ and $47.25$ .

$47.41984164$