Calculus Examples

$\int_{0}^{4} - x^{2} - 3 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{4} - x^{2} d x + \int_{0}^{4} - 3 x d x$

Step 2

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

$- \int_{0}^{4} x^{2} d x + \int_{0}^{4} - 3 x d x$

Step 3

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

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

Step 4

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

$- (\frac{x^{3}}{3}]_{0}^{4}) + \int_{0}^{4} - 3 x d x$

Step 5

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- (\frac{x^{3}}{3}]_{0}^{4}) - 3 \int_{0}^{4} x d x$

Step 6

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

$- (\frac{x^{3}}{3}]_{0}^{4}) - 3 (\frac{1}{2} x^{2}]_{0}^{4})$

Step 7

Simplify the answer.

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Step 7.1

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

$- (\frac{x^{3}}{3}]_{0}^{4}) - 3 (\frac{x^{2}}{2}]_{0}^{4})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $0$ .

$- ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}) - 3 (\frac{x^{2}}{2}]_{0}^{4})$

Step 7.2.2

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

$- (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $4$ to the power of $3$ .

$- (\frac{64}{3} - \frac{0^{3}}{3}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.2

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

$- (\frac{64}{3} - \frac{0}{3}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.3

Cancel the common factor of $0$ and $3$ .

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Step 7.2.3.3.1

Factor $3$ out of $0$ .

$- (\frac{64}{3} - \frac{3 (0)}{3}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $3$ out of $3$ .

$- (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.3.2.2

Cancel the common factor.

$- (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.3.2.3

Rewrite the expression.

$- (\frac{64}{3} - \frac{0}{1}) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.3.2.4

Divide $0$ by $1$ .

$- (\frac{64}{3} - 0) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.4

Multiply $- 1$ by $0$ .

$- (\frac{64}{3} + 0) - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.5

Add $\frac{64}{3}$ and $0$ .

$- \frac{64}{3} - 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 7.2.3.6

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

$- \frac{64}{3} - 3 (\frac{16}{2} - \frac{0^{2}}{2})$

Step 7.2.3.7

Cancel the common factor of $16$ and $2$ .

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Step 7.2.3.7.1

Factor $2$ out of $16$ .

$- \frac{64}{3} - 3 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 7.2.3.7.2

Cancel the common factors.

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Step 7.2.3.7.2.1

Factor $2$ out of $2$ .

$- \frac{64}{3} - 3 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 7.2.3.7.2.2

Cancel the common factor.

$- \frac{64}{3} - 3 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 7.2.3.7.2.3

Rewrite the expression.

$- \frac{64}{3} - 3 (\frac{8}{1} - \frac{0^{2}}{2})$

Step 7.2.3.7.2.4

Divide $8$ by $1$ .

$- \frac{64}{3} - 3 (8 - \frac{0^{2}}{2})$

Step 7.2.3.8

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

$- \frac{64}{3} - 3 (8 - \frac{0}{2})$

Step 7.2.3.9

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

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Step 7.2.3.9.1

Factor $2$ out of $0$ .

$- \frac{64}{3} - 3 (8 - \frac{2 (0)}{2})$

Step 7.2.3.9.2

Cancel the common factors.

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Step 7.2.3.9.2.1

Factor $2$ out of $2$ .

$- \frac{64}{3} - 3 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 7.2.3.9.2.2

Cancel the common factor.

$- \frac{64}{3} - 3 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 7.2.3.9.2.3

Rewrite the expression.

$- \frac{64}{3} - 3 (8 - \frac{0}{1})$

Step 7.2.3.9.2.4

Divide $0$ by $1$ .

$- \frac{64}{3} - 3 (8 - 0)$

Step 7.2.3.10

Multiply $- 1$ by $0$ .

$- \frac{64}{3} - 3 (8 + 0)$

Step 7.2.3.11

Add $8$ and $0$ .

$- \frac{64}{3} - 3 \cdot 8$

Step 7.2.3.12

Multiply $- 3$ by $8$ .

$- \frac{64}{3} - 24$

Step 7.2.3.13

To write $- 24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{64}{3} - 24 \cdot \frac{3}{3}$

Step 7.2.3.14

Combine $- 24$ and $\frac{3}{3}$ .

$- \frac{64}{3} + \frac{- 24 \cdot 3}{3}$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{- 64 - 24 \cdot 3}{3}$

Step 7.2.3.16

Simplify the numerator.

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Step 7.2.3.16.1

Multiply $- 24$ by $3$ .

$\frac{- 64 - 72}{3}$

Step 7.2.3.16.2

Subtract $72$ from $- 64$ .

$\frac{- 136}{3}$

Step 7.2.3.17

Move the negative in front of the fraction.

$- \frac{136}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{136}{3}$

Decimal Form:

$- 45.\overline{3}$

Mixed Number Form:

$- 45 \frac{1}{3}$

Step 9