Calculus Examples

$\int_{1}^{\infty} \frac{1}{x^{3}} d x$

Step 1

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

$lim_{t \to \infty} \int_{1}^{t} \frac{1}{x^{3}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$lim_{t \to \infty} \int_{1}^{t} {(x^{3})}^{- 1} d x$

Step 2.2

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

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Step 2.2.1

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

$lim_{t \to \infty} \int_{1}^{t} x^{3 \cdot - 1} d x$

Step 2.2.2

Multiply $3$ by $- 1$ .

$lim_{t \to \infty} \int_{1}^{t} x^{- 3} d x$

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

One to any power is one.

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

Step 4.2.4

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

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

Step 5

Evaluate the limit.

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Step 5.1

Evaluate the limit.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{t^{2}}$ approaches $0$ .

$- \frac{1}{2} \cdot 0 + lim_{t \to \infty} \frac{1}{2}$

Step 5.3

Evaluate the limit.

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Step 5.3.1

Evaluate the limit of $\frac{1}{2}$ which is constant as $t$ approaches $\infty$ .

$- \frac{1}{2} \cdot 0 + \frac{1}{2}$

Step 5.3.2

Simplify the answer.

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Step 5.3.2.1

Multiply $- \frac{1}{2} \cdot 0$ .

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Step 5.3.2.1.1

Multiply $0$ by $- 1$ .

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

Step 5.3.2.1.2

Multiply $0$ by $\frac{1}{2}$ .

$0 + \frac{1}{2}$

Step 5.3.2.2

Add $0$ and $\frac{1}{2}$ .

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$