Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 10

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

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

Step 11

Simplify the expression.

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Step 11.1

Multiply $2$ by $- 1$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

Multiply $2$ by $- 1$ .

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

Step 12

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

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

Step 13

Substitute and simplify.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

Evaluate $- x^{- 1}$ at $2$ and at $1$ .

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

Step 13.4

Simplify.

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Step 13.4.1

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

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

Step 13.4.2

One to any power is one.

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

Step 13.4.3

Combine the numerators over the common denominator.

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

Step 13.4.4

Subtract $1$ from $8$ .

$3 (\frac{7}{3}) - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.5

Combine $3$ and $\frac{7}{3}$ .

$\frac{3 \cdot 7}{3} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.6

Multiply $3$ by $7$ .

$\frac{21}{3} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.7

Cancel the common factor of $21$ and $3$ .

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Step 13.4.7.1

Factor $3$ out of $21$ .

$\frac{3 \cdot 7}{3} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.7.2

Cancel the common factors.

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Step 13.4.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 7}{3 (1)} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.7.2.2

Cancel the common factor.

$\frac{3 \cdot 7}{3 \cdot 1} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.7.2.3

Rewrite the expression.

$\frac{7}{1} - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.7.2.4

Divide $7$ by $1$ .

$7 - 4 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.8

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

$7 - 4 (\frac{4}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.9

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

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Step 13.4.9.1

Factor $2$ out of $4$ .

$7 - 4 (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.9.2

Cancel the common factors.

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Step 13.4.9.2.1

Factor $2$ out of $2$ .

$7 - 4 (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.9.2.2

Cancel the common factor.

$7 - 4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.9.2.3

Rewrite the expression.

$7 - 4 (\frac{2}{1} - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.9.2.4

Divide $2$ by $1$ .

$7 - 4 (2 - \frac{1^{2}}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.10

One to any power is one.

$7 - 4 (2 - \frac{1}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.11

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

$7 - 4 (2 \cdot \frac{2}{2} - \frac{1}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.12

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

$7 - 4 (\frac{2 \cdot 2}{2} - \frac{1}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.13

Combine the numerators over the common denominator.

$7 - 4 \frac{2 \cdot 2 - 1}{2} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.14

Simplify the numerator.

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Step 13.4.14.1

Multiply $2$ by $2$ .

$7 - 4 \frac{4 - 1}{2} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.14.2

Subtract $1$ from $4$ .

$7 - 4 (\frac{3}{2}) - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.15

Combine $- 4$ and $\frac{3}{2}$ .

$7 + \frac{- 4 \cdot 3}{2} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.16

Multiply $- 4$ by $3$ .

$7 + \frac{- 12}{2} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.17

Cancel the common factor of $- 12$ and $2$ .

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Step 13.4.17.1

Factor $2$ out of $- 12$ .

$7 + \frac{2 \cdot - 6}{2} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.17.2

Cancel the common factors.

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Step 13.4.17.2.1

Factor $2$ out of $2$ .

$7 + \frac{2 \cdot - 6}{2 (1)} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.17.2.2

Cancel the common factor.

$7 + \frac{2 \cdot - 6}{2 \cdot 1} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.17.2.3

Rewrite the expression.

$7 + \frac{- 6}{1} - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.17.2.4

Divide $- 6$ by $1$ .

$7 - 6 - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.18

Subtract $6$ from $7$ .

$1 - 2 ((- 2^{- 1}) + 1^{- 1})$

Step 13.4.19

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

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

Step 13.4.20

One to any power is one.

$1 - 2 (- \frac{1}{2} + 1)$

Step 13.4.21

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

$1 - 2 (- \frac{1}{2} + \frac{2}{2})$

Step 13.4.22

Combine the numerators over the common denominator.

$1 - 2 \frac{- 1 + 2}{2}$

Step 13.4.23

Add $- 1$ and $2$ .

$1 - 2 (\frac{1}{2})$

Step 13.4.24

Combine $- 2$ and $\frac{1}{2}$ .

$1 + \frac{- 2}{2}$

Step 13.4.25

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

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Step 13.4.25.1

Factor $2$ out of $- 2$ .

$1 + \frac{2 \cdot - 1}{2}$

Step 13.4.25.2

Cancel the common factors.

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Step 13.4.25.2.1

Factor $2$ out of $2$ .

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

Step 13.4.25.2.2

Cancel the common factor.

$1 + \frac{2 \cdot - 1}{2 \cdot 1}$

Step 13.4.25.2.3

Rewrite the expression.

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

Step 13.4.25.2.4

Divide $- 1$ by $1$ .

$1 - 1$

Step 13.4.26

Subtract $1$ from $1$ .

$0$

Step 14