Calculus Examples

$\int_{1}^{2} (\frac{12}{x^{2}} - 1) d x$

Step 1

Remove parentheses.

$\int_{1}^{2} \frac{12}{x^{2}} - 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{2} \frac{12}{x^{2}} d x + \int_{1}^{2} - 1 d x$

Step 3

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int_{1}^{2} \frac{1}{x^{2}} d x + \int_{1}^{2} - 1 d x$

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.

$12 \int_{1}^{2} {(x^{2})}^{- 1} d x + \int_{1}^{2} - 1 d x$

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, ${(a^{m})}^{n} = a^{m n}$ .

$12 \int_{1}^{2} x^{2 \cdot - 1} d x + \int_{1}^{2} - 1 d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$12 \int_{1}^{2} x^{- 2} d x + \int_{1}^{2} - 1 d x$

Step 5

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

$12 (- x^{- 1}]_{1}^{2}) + \int_{1}^{2} - 1 d x$

Step 6

Apply the constant rule.

$12 (- x^{- 1}]_{1}^{2}) + - x]_{1}^{2}$

Step 7

Substitute and simplify.

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Step 7.1

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

$12 ((- 2^{- 1}) + 1^{- 1}) + - x]_{1}^{2}$

Step 7.2

Evaluate $- x$ at $2$ and at $1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

One to any power is one.

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

Step 7.3.3

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

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Add $- 1$ and $2$ .

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

Step 7.3.6

Combine $12$ and $\frac{1}{2}$ .

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

Step 7.3.7

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

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Step 7.3.7.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 6}{2} - 1 \cdot 2 + 1 \cdot 1$

Step 7.3.7.2

Cancel the common factors.

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Step 7.3.7.2.1

Factor $2$ out of $2$ .

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

Step 7.3.7.2.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 \cdot 1} - 1 \cdot 2 + 1 \cdot 1$

Step 7.3.7.2.3

Rewrite the expression.

$\frac{6}{1} - 1 \cdot 2 + 1 \cdot 1$

Step 7.3.7.2.4

Divide $6$ by $1$ .

$6 - 1 \cdot 2 + 1 \cdot 1$

Step 7.3.8

Multiply $- 1$ by $2$ .

$6 - 2 + 1 \cdot 1$

Step 7.3.9

Multiply $1$ by $1$ .

$6 - 2 + 1$

Step 7.3.10

Add $- 2$ and $1$ .

$6 - 1$

Step 7.3.11

Subtract $1$ from $6$ .

$5$

Step 8