Calculus Examples

$\int_{1}^{4} (2 x - 3 \sqrt{x}) d x$

Step 1

Remove parentheses.

$\int_{1}^{4} 2 x - 3 \sqrt{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{4} 2 x d x + \int_{1}^{4} - 3 \sqrt{x} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{1}^{4} x d x + \int_{1}^{4} - 3 \sqrt{x} d x$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

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

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $1$ .

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

Step 9.2.2

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $4$ and at $1$ .

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

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

$2 (\frac{16}{2} - \frac{1^{2}}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.2

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

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Step 9.2.3.2.1

Factor $2$ out of $16$ .

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

Step 9.2.3.2.2

Cancel the common factors.

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Step 9.2.3.2.2.1

Factor $2$ out of $2$ .

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

Step 9.2.3.2.2.2

Cancel the common factor.

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

Step 9.2.3.2.2.3

Rewrite the expression.

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

Step 9.2.3.2.2.4

Divide $8$ by $1$ .

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

Step 9.2.3.3

One to any power is one.

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

Step 9.2.3.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9.2.3.5

Combine $8$ and $\frac{2}{2}$ .

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

Step 9.2.3.6

Combine the numerators over the common denominator.

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

Step 9.2.3.7

Simplify the numerator.

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Step 9.2.3.7.1

Multiply $8$ by $2$ .

$2 \frac{16 - 1}{2} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.7.2

Subtract $1$ from $16$ .

$2 (\frac{15}{2}) - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.8

Combine $2$ and $\frac{15}{2}$ .

$\frac{2 \cdot 15}{2} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.9

Multiply $2$ by $15$ .

$\frac{30}{2} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10

Cancel the common factor of $30$ and $2$ .

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Step 9.2.3.10.1

Factor $2$ out of $30$ .

$\frac{2 \cdot 15}{2} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10.2

Cancel the common factors.

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Step 9.2.3.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 15}{2 (1)} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot 15}{2 \cdot 1} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10.2.3

Rewrite the expression.

$\frac{15}{1} - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.10.2.4

Divide $15$ by $1$ .

$15 - 3 (\frac{2 \cdot 4^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.11

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

$15 - 3 (\frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.12

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

$15 - 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.13

Cancel the common factor of $2$ .

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Step 9.2.3.13.1

Cancel the common factor.

$15 - 3 (\frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.13.2

Rewrite the expression.

$15 - 3 (\frac{2 \cdot 2^{3}}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.14

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

$15 - 3 (\frac{2 \cdot 8}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.15

Multiply $2$ by $8$ .

$15 - 3 (\frac{16}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 9.2.3.16

One to any power is one.

$15 - 3 (\frac{16}{3} - \frac{2 \cdot 1}{3})$

Step 9.2.3.17

Multiply $2$ by $1$ .

$15 - 3 (\frac{16}{3} - \frac{2}{3})$

Step 9.2.3.18

Combine the numerators over the common denominator.

$15 - 3 \frac{16 - 2}{3}$

Step 9.2.3.19

Subtract $2$ from $16$ .

$15 - 3 (\frac{14}{3})$

Step 9.2.3.20

Combine $- 3$ and $\frac{14}{3}$ .

$15 + \frac{- 3 \cdot 14}{3}$

Step 9.2.3.21

Multiply $- 3$ by $14$ .

$15 + \frac{- 42}{3}$

Step 9.2.3.22

Cancel the common factor of $- 42$ and $3$ .

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Step 9.2.3.22.1

Factor $3$ out of $- 42$ .

$15 + \frac{3 \cdot - 14}{3}$

Step 9.2.3.22.2

Cancel the common factors.

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Step 9.2.3.22.2.1

Factor $3$ out of $3$ .

$15 + \frac{3 \cdot - 14}{3 (1)}$

Step 9.2.3.22.2.2

Cancel the common factor.

$15 + \frac{3 \cdot - 14}{3 \cdot 1}$

Step 9.2.3.22.2.3

Rewrite the expression.

$15 + \frac{- 14}{1}$

Step 9.2.3.22.2.4

Divide $- 14$ by $1$ .

$15 - 14$

Step 9.2.3.23

Subtract $14$ from $15$ .

$1$

Step 10