Calculus Examples

$\int_{0}^{5} (975 - 23 x) - 630 d x$

Step 1

Subtract $630$ from $975$ .

$\int_{0}^{5} - 23 x + 345 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{5} - 23 x d x + \int_{0}^{5} 345 d x$

Step 3

Since $- 23$ is constant with respect to $x$ , move $- 23$ out of the integral.

$- 23 \int_{0}^{5} x d x + \int_{0}^{5} 345 d x$

Step 4

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

$- 23 (\frac{1}{2} x^{2}]_{0}^{5}) + \int_{0}^{5} 345 d x$

Step 5

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

$- 23 (\frac{x^{2}}{2}]_{0}^{5}) + \int_{0}^{5} 345 d x$

Step 6

Apply the constant rule.

$- 23 (\frac{x^{2}}{2}]_{0}^{5}) + 345 x]_{0}^{5}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $0$ .

$- 23 ((\frac{5^{2}}{2}) - \frac{0^{2}}{2}) + 345 x]_{0}^{5}$

Step 7.2

Evaluate $345 x$ at $5$ and at $0$ .

$- 23 (\frac{5^{2}}{2} - \frac{0^{2}}{2}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3

Simplify.

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Step 7.3.1

Raise $5$ to the power of $2$ .

$- 23 (\frac{25}{2} - \frac{0^{2}}{2}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.2

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

$- 23 (\frac{25}{2} - \frac{0}{2}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.3

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

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Step 7.3.3.1

Factor $2$ out of $0$ .

$- 23 (\frac{25}{2} - \frac{2 (0)}{2}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.3.2

Cancel the common factors.

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Step 7.3.3.2.1

Factor $2$ out of $2$ .

$- 23 (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.3.2.2

Cancel the common factor.

$- 23 (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.3.2.3

Rewrite the expression.

$- 23 (\frac{25}{2} - \frac{0}{1}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.3.2.4

Divide $0$ by $1$ .

$- 23 (\frac{25}{2} - 0) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.4

Multiply $- 1$ by $0$ .

$- 23 (\frac{25}{2} + 0) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.5

Add $\frac{25}{2}$ and $0$ .

$- 23 (\frac{25}{2}) + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.6

Combine $- 23$ and $\frac{25}{2}$ .

$\frac{- 23 \cdot 25}{2} + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.7

Multiply $- 23$ by $25$ .

$\frac{- 575}{2} + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.8

Move the negative in front of the fraction.

$- \frac{575}{2} + 345 \cdot 5 - 345 \cdot 0$

Step 7.3.9

Multiply $345$ by $5$ .

$- \frac{575}{2} + 1725 - 345 \cdot 0$

Step 7.3.10

Multiply $- 345$ by $0$ .

$- \frac{575}{2} + 1725 + 0$

Step 7.3.11

Add $1725$ and $0$ .

$- \frac{575}{2} + 1725$

Step 7.3.12

To write $1725$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{575}{2} + 1725 \cdot \frac{2}{2}$

Step 7.3.13

Combine $1725$ and $\frac{2}{2}$ .

$- \frac{575}{2} + \frac{1725 \cdot 2}{2}$

Step 7.3.14

Combine the numerators over the common denominator.

$\frac{- 575 + 1725 \cdot 2}{2}$

Step 7.3.15

Simplify the numerator.

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Step 7.3.15.1

Multiply $1725$ by $2$ .

$\frac{- 575 + 3450}{2}$

Step 7.3.15.2

Add $- 575$ and $3450$ .

$\frac{2875}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2875}{2}$

Decimal Form:

$1437.5$

Mixed Number Form:

$1437 \frac{1}{2}$

Step 9