Calculus Examples

$\int_{1}^{\infty} \frac{5}{x^{4}} d x$

Step 1

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

$lim_{t \to \infty} \int_{1}^{t} \frac{5}{x^{4}} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$lim_{t \to \infty} 5 \int_{1}^{t} \frac{1}{x^{4}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$lim_{t \to \infty} 5 \int_{1}^{t} {(x^{4})}^{- 1} d x$

Step 3.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 3.2.1

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

$lim_{t \to \infty} 5 \int_{1}^{t} x^{4 \cdot - 1} d x$

Step 3.2.2

Multiply $4$ by $- 1$ .

$lim_{t \to \infty} 5 \int_{1}^{t} x^{- 4} d x$

Step 4

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$lim_{t \to \infty} 5 (- \frac{1}{3} x^{- 3}]_{1}^{t})$

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

$lim_{t \to \infty} 5 (- \frac{x^{- 3}}{3}]_{1}^{t})$

Step 5.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to \infty} 5 (- \frac{1}{3 x^{3}}]_{1}^{t})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \frac{1}{3 x^{3}}$ at $t$ and at $1$ .

$lim_{t \to \infty} 5 ((- \frac{1}{3 t^{3}}) + \frac{1}{3 \cdot 1^{3}})$

Step 5.2.2

Simplify.

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Step 5.2.2.1

One to any power is one.

$lim_{t \to \infty} 5 (- \frac{1}{3 t^{3}} + \frac{1}{3 \cdot 1})$

Step 5.2.2.2

Multiply $3$ by $1$ .

$lim_{t \to \infty} 5 (- \frac{1}{3 t^{3}} + \frac{1}{3})$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit.

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Step 6.1.1

Move the term $5$ outside of the limit because it is constant with respect to $t$ .

$5 lim_{t \to \infty} - \frac{1}{3 t^{3}} + \frac{1}{3}$

Step 6.1.2

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

$5 (- lim_{t \to \infty} \frac{1}{3 t^{3}} + lim_{t \to \infty} \frac{1}{3})$

Step 6.1.3

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $t$ .

$5 (- \frac{1}{3} lim_{t \to \infty} \frac{1}{t^{3}} + lim_{t \to \infty} \frac{1}{3})$

Step 6.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{t^{3}}$ approaches $0$ .

$5 (- \frac{1}{3} \cdot 0 + lim_{t \to \infty} \frac{1}{3})$

Step 6.3

Evaluate the limit.

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Step 6.3.1

Evaluate the limit of $\frac{1}{3}$ which is constant as $t$ approaches $\infty$ .

$5 (- \frac{1}{3} \cdot 0 + \frac{1}{3})$

Step 6.3.2

Simplify the answer.

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Step 6.3.2.1

Multiply $- \frac{1}{3} \cdot 0$ .

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Step 6.3.2.1.1

Multiply $0$ by $- 1$ .

$5 (0 (\frac{1}{3}) + \frac{1}{3})$

Step 6.3.2.1.2

Multiply $0$ by $\frac{1}{3}$ .

$5 (0 + \frac{1}{3})$

Step 6.3.2.2

Add $0$ and $\frac{1}{3}$ .

$5 (\frac{1}{3})$

Step 6.3.2.3

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{3}$

Decimal Form:

$1.\overline{6}$

Mixed Number Form:

$1 \frac{2}{3}$