Calculus Examples

$\int_{0}^{2} 9 x^{2} - 8 x + 1 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{2} 9 x^{2} d x + \int_{0}^{2} - 8 x d x + \int_{0}^{2} d x$

Step 2

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{0}^{2} x^{2} d x + \int_{0}^{2} - 8 x d x + \int_{0}^{2} d x$

Step 3

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

$9 (\frac{1}{3} x^{3}]_{0}^{2}) + \int_{0}^{2} - 8 x d x + \int_{0}^{2} d x$

Step 4

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

$9 (\frac{x^{3}}{3}]_{0}^{2}) + \int_{0}^{2} - 8 x d x + \int_{0}^{2} d x$

Step 5

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$9 (\frac{x^{3}}{3}]_{0}^{2}) - 8 \int_{0}^{2} x d x + \int_{0}^{2} d x$

Step 6

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

$9 (\frac{x^{3}}{3}]_{0}^{2}) - 8 (\frac{1}{2} x^{2}]_{0}^{2}) + \int_{0}^{2} d x$

Step 7

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

$9 (\frac{x^{3}}{3}]_{0}^{2}) - 8 (\frac{x^{2}}{2}]_{0}^{2}) + \int_{0}^{2} d x$

Step 8

Apply the constant rule.

$9 (\frac{x^{3}}{3}]_{0}^{2}) - 8 (\frac{x^{2}}{2}]_{0}^{2}) + x]_{0}^{2}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

$9 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3}) - 8 (\frac{x^{2}}{2}]_{0}^{2}) + x]_{0}^{2}$

Step 9.2

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

$9 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + x]_{0}^{2}$

Step 9.3

Evaluate $x$ at $2$ and at $0$ .

$9 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4

Simplify.

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Step 9.4.1

Raise $2$ to the power of $3$ .

$9 (\frac{8}{3} - \frac{0^{3}}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.2

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

$9 (\frac{8}{3} - \frac{0}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.3

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

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Step 9.4.3.1

Factor $3$ out of $0$ .

$9 (\frac{8}{3} - \frac{3 (0)}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.3.2

Cancel the common factors.

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Step 9.4.3.2.1

Factor $3$ out of $3$ .

$9 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.3.2.2

Cancel the common factor.

$9 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.3.2.3

Rewrite the expression.

$9 (\frac{8}{3} - \frac{0}{1}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.3.2.4

Divide $0$ by $1$ .

$9 (\frac{8}{3} - 0) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.4

Multiply $- 1$ by $0$ .

$9 (\frac{8}{3} + 0) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.5

Add $\frac{8}{3}$ and $0$ .

$9 (\frac{8}{3}) - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.6

Combine $9$ and $\frac{8}{3}$ .

$\frac{9 \cdot 8}{3} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.7

Multiply $9$ by $8$ .

$\frac{72}{3} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.8

Cancel the common factor of $72$ and $3$ .

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Step 9.4.8.1

Factor $3$ out of $72$ .

$\frac{3 \cdot 24}{3} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.8.2

Cancel the common factors.

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Step 9.4.8.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 24}{3 (1)} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.8.2.2

Cancel the common factor.

$\frac{3 \cdot 24}{3 \cdot 1} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.8.2.3

Rewrite the expression.

$\frac{24}{1} - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.8.2.4

Divide $24$ by $1$ .

$24 - 8 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.9

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

$24 - 8 (\frac{4}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.10

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

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Step 9.4.10.1

Factor $2$ out of $4$ .

$24 - 8 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.10.2

Cancel the common factors.

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Step 9.4.10.2.1

Factor $2$ out of $2$ .

$24 - 8 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.10.2.2

Cancel the common factor.

$24 - 8 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.10.2.3

Rewrite the expression.

$24 - 8 (\frac{2}{1} - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.10.2.4

Divide $2$ by $1$ .

$24 - 8 (2 - \frac{0^{2}}{2}) + (2) + 0$

Step 9.4.11

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

$24 - 8 (2 - \frac{0}{2}) + (2) + 0$

Step 9.4.12

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

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Step 9.4.12.1

Factor $2$ out of $0$ .

$24 - 8 (2 - \frac{2 (0)}{2}) + (2) + 0$

Step 9.4.12.2

Cancel the common factors.

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Step 9.4.12.2.1

Factor $2$ out of $2$ .

$24 - 8 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + (2) + 0$

Step 9.4.12.2.2

Cancel the common factor.

$24 - 8 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + (2) + 0$

Step 9.4.12.2.3

Rewrite the expression.

$24 - 8 (2 - \frac{0}{1}) + (2) + 0$

Step 9.4.12.2.4

Divide $0$ by $1$ .

$24 - 8 (2 - 0) + (2) + 0$

Step 9.4.13

Multiply $- 1$ by $0$ .

$24 - 8 (2 + 0) + (2) + 0$

Step 9.4.14

Add $2$ and $0$ .

$24 - 8 \cdot 2 + (2) + 0$

Step 9.4.15

Multiply $- 8$ by $2$ .

$24 - 16 + (2) + 0$

Step 9.4.16

Subtract $16$ from $24$ .

$8 + (2) + 0$

Step 9.4.17

Add $2$ and $0$ .

$8 + 2$

Step 9.4.18

Add $8$ and $2$ .

$10$

Step 10