Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $- x^{- 1}$ at $4$ and at $1$ .

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

Step 11.2.2

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

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

Step 11.2.3

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

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

Step 11.2.4

Simplify.

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Step 11.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.2.4.2

One to any power is one.

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

Step 11.2.4.3

Write $1$ as a fraction with a common denominator.

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

Step 11.2.4.4

Combine the numerators over the common denominator.

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

Step 11.2.4.5

Add $- 1$ and $4$ .

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

Step 11.2.4.6

Combine $4$ and $\frac{3}{4}$ .

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

Step 11.2.4.7

Multiply $4$ by $3$ .

$\frac{12}{4} + 2 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) - 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 11.2.4.8

Cancel the common factor of $12$ and $4$ .

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Step 11.2.4.8.1

Factor $4$ out of $12$ .

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

Step 11.2.4.8.2

Cancel the common factors.

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Step 11.2.4.8.2.1

Factor $4$ out of $4$ .

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

Step 11.2.4.8.2.2

Cancel the common factor.

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

Step 11.2.4.8.2.3

Rewrite the expression.

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

Step 11.2.4.8.2.4

Divide $3$ by $1$ .

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

Step 11.2.4.9

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

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

Step 11.2.4.10

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

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Step 11.2.4.10.1

Factor $2$ out of $16$ .

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

Step 11.2.4.10.2

Cancel the common factors.

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Step 11.2.4.10.2.1

Factor $2$ out of $2$ .

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

Step 11.2.4.10.2.2

Cancel the common factor.

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

Step 11.2.4.10.2.3

Rewrite the expression.

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

Step 11.2.4.10.2.4

Divide $8$ by $1$ .

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

Step 11.2.4.11

One to any power is one.

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

Step 11.2.4.12

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

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

Step 11.2.4.13

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

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

Step 11.2.4.14

Combine the numerators over the common denominator.

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

Step 11.2.4.15

Simplify the numerator.

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Step 11.2.4.15.1

Multiply $8$ by $2$ .

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

Step 11.2.4.15.2

Subtract $1$ from $16$ .

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

Step 11.2.4.16

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

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

Step 11.2.4.17

Multiply $2$ by $15$ .

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

Step 11.2.4.18

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

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Step 11.2.4.18.1

Factor $2$ out of $30$ .

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

Step 11.2.4.18.2

Cancel the common factors.

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Step 11.2.4.18.2.1

Factor $2$ out of $2$ .

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

Step 11.2.4.18.2.2

Cancel the common factor.

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

Step 11.2.4.18.2.3

Rewrite the expression.

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

Step 11.2.4.18.2.4

Divide $15$ by $1$ .

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

Step 11.2.4.19

Add $3$ and $15$ .

$18 - 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 11.2.4.20

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

$18 - 3 (\frac{64}{3} - \frac{1^{3}}{3})$

Step 11.2.4.21

One to any power is one.

$18 - 3 (\frac{64}{3} - \frac{1}{3})$

Step 11.2.4.22

Combine the numerators over the common denominator.

$18 - 3 \frac{64 - 1}{3}$

Step 11.2.4.23

Subtract $1$ from $64$ .

$18 - 3 (\frac{63}{3})$

Step 11.2.4.24

Cancel the common factor of $63$ and $3$ .

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Step 11.2.4.24.1

Factor $3$ out of $63$ .

$18 - 3 \frac{3 \cdot 21}{3}$

Step 11.2.4.24.2

Cancel the common factors.

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Step 11.2.4.24.2.1

Factor $3$ out of $3$ .

$18 - 3 \frac{3 \cdot 21}{3 (1)}$

Step 11.2.4.24.2.2

Cancel the common factor.

$18 - 3 \frac{3 \cdot 21}{3 \cdot 1}$

Step 11.2.4.24.2.3

Rewrite the expression.

$18 - 3 (\frac{21}{1})$

Step 11.2.4.24.2.4

Divide $21$ by $1$ .

$18 - 3 \cdot 21$

Step 11.2.4.25

Multiply $- 3$ by $21$ .

$18 - 63$

Step 11.2.4.26

Subtract $63$ from $18$ .

$- 45$

Step 12