Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $4$ and at $0$ .

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

$4 (\frac{16}{2} - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.2

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

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Step 8.2.3.2.1

Factor $2$ out of $16$ .

$4 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.2.2.2

Cancel the common factor.

$4 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.2.2.3

Rewrite the expression.

$4 (\frac{8}{1} - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.2.2.4

Divide $8$ by $1$ .

$4 (8 - \frac{0^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.3

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

$4 (8 - \frac{0}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.4

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

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Step 8.2.3.4.1

Factor $2$ out of $0$ .

$4 (8 - \frac{2 (0)}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.4.2

Cancel the common factors.

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Step 8.2.3.4.2.1

Factor $2$ out of $2$ .

$4 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.4.2.2

Cancel the common factor.

$4 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.4.2.3

Rewrite the expression.

$4 (8 - \frac{0}{1}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.4.2.4

Divide $0$ by $1$ .

$4 (8 - 0) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.5

Multiply $- 1$ by $0$ .

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

Step 8.2.3.6

Add $8$ and $0$ .

$4 \cdot 8 - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.7

Multiply $4$ by $8$ .

$32 - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.8

Rewrite $4$ as $2^{2}$ .

$32 - 3 (\frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.9

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$32 - 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.10

Cancel the common factor of $2$ .

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Step 8.2.3.10.1

Cancel the common factor.

$32 - 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.10.2

Rewrite the expression.

$32 - 3 (\frac{2 \cdot 2^{3}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.11

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

$32 - 3 (\frac{2 \cdot 8}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.12

Multiply $2$ by $8$ .

$32 - 3 (\frac{16}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 8.2.3.13

Rewrite $0$ as $0^{2}$ .

$32 - 3 (\frac{16}{3} - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3})$

Step 8.2.3.14

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$32 - 3 (\frac{16}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 8.2.3.15

Cancel the common factor of $2$ .

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Step 8.2.3.15.1

Cancel the common factor.

$32 - 3 (\frac{16}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 8.2.3.15.2

Rewrite the expression.

$32 - 3 (\frac{16}{3} - \frac{2 \cdot 0^{3}}{3})$

Step 8.2.3.16

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

$32 - 3 (\frac{16}{3} - \frac{2 \cdot 0}{3})$

Step 8.2.3.17

Multiply $2$ by $0$ .

$32 - 3 (\frac{16}{3} - \frac{0}{3})$

Step 8.2.3.18

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

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Step 8.2.3.18.1

Factor $3$ out of $0$ .

$32 - 3 (\frac{16}{3} - \frac{3 (0)}{3})$

Step 8.2.3.18.2

Cancel the common factors.

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Step 8.2.3.18.2.1

Factor $3$ out of $3$ .

$32 - 3 (\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.18.2.2

Cancel the common factor.

$32 - 3 (\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.18.2.3

Rewrite the expression.

$32 - 3 (\frac{16}{3} - \frac{0}{1})$

Step 8.2.3.18.2.4

Divide $0$ by $1$ .

$32 - 3 (\frac{16}{3} - 0)$

Step 8.2.3.19

Multiply $- 1$ by $0$ .

$32 - 3 (\frac{16}{3} + 0)$

Step 8.2.3.20

Add $\frac{16}{3}$ and $0$ .

$32 - 3 (\frac{16}{3})$

Step 8.2.3.21

Combine $- 3$ and $\frac{16}{3}$ .

$32 + \frac{- 3 \cdot 16}{3}$

Step 8.2.3.22

Multiply $- 3$ by $16$ .

$32 + \frac{- 48}{3}$

Step 8.2.3.23

Cancel the common factor of $- 48$ and $3$ .

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Step 8.2.3.23.1

Factor $3$ out of $- 48$ .

$32 + \frac{3 \cdot - 16}{3}$

Step 8.2.3.23.2

Cancel the common factors.

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Step 8.2.3.23.2.1

Factor $3$ out of $3$ .

$32 + \frac{3 \cdot - 16}{3 (1)}$

Step 8.2.3.23.2.2

Cancel the common factor.

$32 + \frac{3 \cdot - 16}{3 \cdot 1}$

Step 8.2.3.23.2.3

Rewrite the expression.

$32 + \frac{- 16}{1}$

Step 8.2.3.23.2.4

Divide $- 16$ by $1$ .

$32 - 16$

Step 8.2.3.24

Subtract $16$ from $32$ .

$16$

Step 9