Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Expand ${(3 x^{2} + 2)}^{2}$ .

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Step 2.1

Rewrite ${(3 x^{2} + 2)}^{2}$ as $(3 x^{2} + 2) (3 x^{2} + 2)$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$π \int_{0}^{2} 3 x^{2} (3 x^{2}) + 3 x^{2} \cdot 2 + 2 (3 x^{2} + 2) d x$

Step 2.4

Apply the distributive property.

$π \int_{0}^{2} 3 x^{2} (3 x^{2}) + 3 x^{2} \cdot 2 + 2 (3 x^{2}) + 2 \cdot 2 d x$

Step 2.5

Move $x^{2}$ .

$π \int_{0}^{2} 3 \cdot 3 x^{2} x^{2} + 3 x^{2} \cdot 2 + 2 (3 x^{2}) + 2 \cdot 2 d x$

Step 2.6

Move $x^{2}$ .

$π \int_{0}^{2} 3 \cdot 3 x^{2} x^{2} + 3 \cdot 2 x^{2} + 2 (3 x^{2}) + 2 \cdot 2 d x$

Step 2.7

Multiply $3$ by $3$ .

$π \int_{0}^{2} 9 x^{2} x^{2} + 3 \cdot 2 x^{2} + 2 \cdot 3 x^{2} + 2 \cdot 2 d x$

Step 2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$π \int_{0}^{2} 9 x^{2 + 2} + 3 \cdot 2 x^{2} + 2 \cdot 3 x^{2} + 2 \cdot 2 d x$

Step 2.9

Add $2$ and $2$ .

$π \int_{0}^{2} 9 x^{4} + 3 \cdot 2 x^{2} + 2 \cdot 3 x^{2} + 2 \cdot 2 d x$

Step 2.10

Multiply $3$ by $2$ .

$π \int_{0}^{2} 9 x^{4} + 6 x^{2} + 2 \cdot 3 x^{2} + 2 \cdot 2 d x$

Step 2.11

Multiply $2$ by $3$ .

$π \int_{0}^{2} 9 x^{4} + 6 x^{2} + 6 x^{2} + 2 \cdot 2 d x$

Step 2.12

Multiply $2$ by $2$ .

$π \int_{0}^{2} 9 x^{4} + 6 x^{2} + 6 x^{2} + 4 d x$

Step 2.13

Add $6 x^{2}$ and $6 x^{2}$ .

$π \int_{0}^{2} 9 x^{4} + 12 x^{2} + 4 d x$

Step 3

Split the single integral into multiple integrals.

$π (\int_{0}^{2} 9 x^{4} d x + \int_{0}^{2} 12 x^{2} d x + \int_{0}^{2} 4 d x)$

Step 4

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

$π (9 \int_{0}^{2} x^{4} d x + \int_{0}^{2} 12 x^{2} d x + \int_{0}^{2} 4 d x)$

Step 5

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$π (9 (\frac{1}{5} x^{5}]_{0}^{2}) + \int_{0}^{2} 12 x^{2} d x + \int_{0}^{2} 4 d x)$

Step 6

Combine $\frac{1}{5}$ and $x^{5}$ .

$π (9 (\frac{x^{5}}{5}]_{0}^{2}) + \int_{0}^{2} 12 x^{2} d x + \int_{0}^{2} 4 d x)$

Step 7

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$π (9 (\frac{x^{5}}{5}]_{0}^{2}) + 12 \int_{0}^{2} x^{2} d x + \int_{0}^{2} 4 d x)$

Step 8

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

$π (9 (\frac{x^{5}}{5}]_{0}^{2}) + 12 (\frac{1}{3} x^{3}]_{0}^{2}) + \int_{0}^{2} 4 d x)$

Step 9

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

$π (9 (\frac{x^{5}}{5}]_{0}^{2}) + 12 (\frac{x^{3}}{3}]_{0}^{2}) + \int_{0}^{2} 4 d x)$

Step 10

Apply the constant rule.

$π (9 (\frac{x^{5}}{5}]_{0}^{2}) + 12 (\frac{x^{3}}{3}]_{0}^{2}) + 4 x]_{0}^{2})$

Step 11

Substitute and simplify.

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Step 11.1

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

$π (9 ((\frac{2^{5}}{5}) - \frac{0^{5}}{5}) + 12 (\frac{x^{3}}{3}]_{0}^{2}) + 4 x]_{0}^{2})$

Step 11.2

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

$π (9 (\frac{2^{5}}{5} - \frac{0^{5}}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + 4 x]_{0}^{2})$

Step 11.3

Evaluate $4 x$ at $2$ and at $0$ .

$π (9 (\frac{2^{5}}{5} - \frac{0^{5}}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4

Simplify.

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Step 11.4.1

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

$π (9 (\frac{32}{5} - \frac{0^{5}}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.2

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

$π (9 (\frac{32}{5} - \frac{0}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.3

Cancel the common factor of $0$ and $5$ .

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Step 11.4.3.1

Factor $5$ out of $0$ .

$π (9 (\frac{32}{5} - \frac{5 (0)}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.3.2

Cancel the common factors.

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Step 11.4.3.2.1

Factor $5$ out of $5$ .

$π (9 (\frac{32}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.3.2.2

Cancel the common factor.

$π (9 (\frac{32}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.3.2.3

Rewrite the expression.

$π (9 (\frac{32}{5} - \frac{0}{1}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.3.2.4

Divide $0$ by $1$ .

$π (9 (\frac{32}{5} - 0) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.4

Multiply $- 1$ by $0$ .

$π (9 (\frac{32}{5} + 0) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.5

Add $\frac{32}{5}$ and $0$ .

$π (9 (\frac{32}{5}) + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.6

Combine $9$ and $\frac{32}{5}$ .

$π (\frac{9 \cdot 32}{5} + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.7

Multiply $9$ by $32$ .

$π (\frac{288}{5} + 12 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.8

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

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{0^{3}}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.9

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

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{0}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.10

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

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Step 11.4.10.1

Factor $3$ out of $0$ .

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{3 (0)}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.10.2

Cancel the common factors.

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Step 11.4.10.2.1

Factor $3$ out of $3$ .

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.10.2.2

Cancel the common factor.

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.10.2.3

Rewrite the expression.

$π (\frac{288}{5} + 12 (\frac{8}{3} - \frac{0}{1}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.10.2.4

Divide $0$ by $1$ .

$π (\frac{288}{5} + 12 (\frac{8}{3} - 0) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.11

Multiply $- 1$ by $0$ .

$π (\frac{288}{5} + 12 (\frac{8}{3} + 0) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.12

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

$π (\frac{288}{5} + 12 (\frac{8}{3}) + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.13

Combine $12$ and $\frac{8}{3}$ .

$π (\frac{288}{5} + \frac{12 \cdot 8}{3} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.14

Multiply $12$ by $8$ .

$π (\frac{288}{5} + \frac{96}{3} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.15

Cancel the common factor of $96$ and $3$ .

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Step 11.4.15.1

Factor $3$ out of $96$ .

$π (\frac{288}{5} + \frac{3 \cdot 32}{3} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.15.2

Cancel the common factors.

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Step 11.4.15.2.1

Factor $3$ out of $3$ .

$π (\frac{288}{5} + \frac{3 \cdot 32}{3 (1)} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.15.2.2

Cancel the common factor.

$π (\frac{288}{5} + \frac{3 \cdot 32}{3 \cdot 1} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.15.2.3

Rewrite the expression.

$π (\frac{288}{5} + \frac{32}{1} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.15.2.4

Divide $32$ by $1$ .

$π (\frac{288}{5} + 32 + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.16

To write $32$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$π (\frac{288}{5} + 32 \cdot \frac{5}{5} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.17

Combine $32$ and $\frac{5}{5}$ .

$π (\frac{288}{5} + \frac{32 \cdot 5}{5} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.18

Combine the numerators over the common denominator.

$π (\frac{288 + 32 \cdot 5}{5} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.19

Simplify the numerator.

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Step 11.4.19.1

Multiply $32$ by $5$ .

$π (\frac{288 + 160}{5} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.19.2

Add $288$ and $160$ .

$π (\frac{448}{5} + (4 \cdot 2) - 4 \cdot 0)$

Step 11.4.20

Multiply $4$ by $2$ .

$π (\frac{448}{5} + 8 - 4 \cdot 0)$

Step 11.4.21

Multiply $- 4$ by $0$ .

$π (\frac{448}{5} + 8 + 0)$

Step 11.4.22

Add $8$ and $0$ .

$π (\frac{448}{5} + 8)$

Step 11.4.23

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$π (\frac{448}{5} + 8 \cdot \frac{5}{5})$

Step 11.4.24

Combine $8$ and $\frac{5}{5}$ .

$π (\frac{448}{5} + \frac{8 \cdot 5}{5})$

Step 11.4.25

Combine the numerators over the common denominator.

$π \frac{448 + 8 \cdot 5}{5}$

Step 11.4.26

Simplify the numerator.

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Step 11.4.26.1

Multiply $8$ by $5$ .

$π \frac{448 + 40}{5}$

Step 11.4.26.2

Add $448$ and $40$ .

$π \frac{488}{5}$

Step 11.4.27

Combine $π$ and $\frac{488}{5}$ .

$\frac{π \cdot 488}{5}$

Step 11.4.28

Move $488$ to the left of $π$ .

$\frac{488 π}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{488 π}{5}$

Decimal Form:

$306.61944299 \dots$

Step 13