Calculus Examples

$\int_{1}^{\infty} \frac{1}{2 x^{2}} d x$

Step 1

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{1}^{t} \frac{1}{2 x^{2}} d x$

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$lim_{t \to \infty} \frac{1}{2} \int_{1}^{t} \frac{1}{x^{2}} d x$

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.

$lim_{t \to \infty} \frac{1}{2} \int_{1}^{t} {(x^{2})}^{- 1} d x$

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

$lim_{t \to \infty} \frac{1}{2} \int_{1}^{t} x^{2 \cdot - 1} d x$

Step 3.2.2

Multiply $2$ by $- 1$ .

$lim_{t \to \infty} \frac{1}{2} \int_{1}^{t} x^{- 2} d x$

Step 4

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

$lim_{t \to \infty} \frac{1}{2} - x^{- 1}]_{1}^{t}$

Step 5

Simplify the answer.

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Step 5.1

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

$lim_{t \to \infty} \frac{- x^{- 1}]_{1}^{t}}{2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

$lim_{t \to \infty} \frac{(- t^{- 1}) + 1^{- 1}}{2}$

Step 5.2.2

One to any power is one.

$lim_{t \to \infty} \frac{- t^{- 1} + 1}{2}$

Step 5.3

Simplify.

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Step 5.3.1

Factor $- 1$ out of $- t^{- 1}$ .

$lim_{t \to \infty} \frac{- (t^{- 1}) + 1}{2}$

Step 5.3.2

Rewrite $1$ as $- 1 (- 1)$ .

$lim_{t \to \infty} \frac{- (t^{- 1}) - 1 (- 1)}{2}$

Step 5.3.3

Factor $- 1$ out of $- (t^{- 1}) - 1 (- 1)$ .

$lim_{t \to \infty} \frac{- (t^{- 1} - 1)}{2}$

Step 5.3.4

Rewrite $- (t^{- 1} - 1)$ as $- 1 (t^{- 1} - 1)$ .

$lim_{t \to \infty} \frac{- 1 (t^{- 1} - 1)}{2}$

Step 5.3.5

Move the negative in front of the fraction.

$lim_{t \to \infty} - \frac{t^{- 1} - 1}{2}$

Step 6

Evaluate the limit.

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Step 6.1

Move the term $- 1$ outside of the limit because it is constant with respect to $t$ .

$- lim_{t \to \infty} \frac{t^{- 1} - 1}{2}$

Step 6.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$- \frac{1}{2} lim_{t \to \infty} t^{- 1} - 1$

Step 6.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\infty$ .

$- \frac{1}{2} (lim_{t \to \infty} t^{- 1} - lim_{t \to \infty} 1)$

Step 6.4

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

$- \frac{1}{2} (lim_{t \to \infty} \frac{1}{t} - lim_{t \to \infty} 1)$

Step 6.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{t}$ approaches $0$ .

$- \frac{1}{2} (0 - lim_{t \to \infty} 1)$

Step 6.6

Evaluate the limit.

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Step 6.6.1

Evaluate the limit of $1$ which is constant as $t$ approaches $\infty$ .

$- \frac{1}{2} (0 - 1 \cdot 1)$

Step 6.6.2

Simplify the answer.

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Step 6.6.2.1

Multiply $- 1$ by $1$ .

$- \frac{1}{2} (0 - 1)$

Step 6.6.2.2

Subtract $1$ from $0$ .

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

Step 6.6.2.3

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

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Step 6.6.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.6.2.3.2

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

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$