Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

Multiply $4$ by $- 1$ .

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

Step 7.2

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.3

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 7.3.1

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

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

Step 7.3.2

Multiply $3$ by $- 1$ .

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

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

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

Step 9.2.3.2

One to any power is one.

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

Step 9.2.3.3

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

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

Step 9.2.3.4

Combine the numerators over the common denominator.

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

Step 9.2.3.5

Add $- 1$ and $2$ .

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

Step 9.2.3.6

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

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

Step 9.2.3.7

Multiply $2$ by $4$ .

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

Step 9.2.3.8

One to any power is one.

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

Step 9.2.3.9

Multiply $2$ by $1$ .

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

Step 9.2.3.10

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 9.2.3.11

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.2.3.11.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

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

Step 9.2.3.11.2

Multiply $2$ by $4$ .

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

Step 9.2.3.12

Combine the numerators over the common denominator.

$\frac{1}{2} - 4 \frac{- 1 + 4}{8}$

Step 9.2.3.13

Add $- 1$ and $4$ .

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

Step 9.2.3.14

Combine $- 4$ and $\frac{3}{8}$ .

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

Step 9.2.3.15

Multiply $- 4$ by $3$ .

$\frac{1}{2} + \frac{- 12}{8}$

Step 9.2.3.16

Cancel the common factor of $- 12$ and $8$ .

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Step 9.2.3.16.1

Factor $4$ out of $- 12$ .

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

Step 9.2.3.16.2

Cancel the common factors.

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Step 9.2.3.16.2.1

Factor $4$ out of $8$ .

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

Step 9.2.3.16.2.2

Cancel the common factor.

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

Step 9.2.3.16.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{- 3}{2}$

Step 9.2.3.17

Move the negative in front of the fraction.

$\frac{1}{2} - \frac{3}{2}$

Step 9.2.3.18

Combine the numerators over the common denominator.

$\frac{1 - 3}{2}$

Step 9.2.3.19

Subtract $3$ from $1$ .

$\frac{- 2}{2}$

Step 9.2.3.20

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

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Step 9.2.3.20.1

Factor $2$ out of $- 2$ .

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

Step 9.2.3.20.2

Cancel the common factors.

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Step 9.2.3.20.2.1

Factor $2$ out of $2$ .

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

Step 9.2.3.20.2.2

Cancel the common factor.

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

Step 9.2.3.20.2.3

Rewrite the expression.

$\frac{- 1}{1}$

Step 9.2.3.20.2.4

Divide $- 1$ by $1$ .

$- 1$

Step 10