Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

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.

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

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

One to any power is one.

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

Step 7.3.3

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

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Add $- 1$ and $2$ .

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

Step 7.3.6

Combine $3$ and $\frac{1}{2}$ .

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

Step 7.3.7

Multiply $- 1$ by $2$ .

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

Step 7.3.8

Multiply $1$ by $1$ .

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

Step 7.3.9

Add $- 2$ and $1$ .

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

Step 7.3.10

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

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

Step 7.3.11

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

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

Step 7.3.12

Combine the numerators over the common denominator.

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

Step 7.3.13

Simplify the numerator.

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Step 7.3.13.1

Multiply $- 1$ by $2$ .

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

Step 7.3.13.2

Subtract $2$ from $3$ .

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 9