Calculus Examples

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

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $4$ by $- 1$ .

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $- \frac{1}{3} x^{- 3}$ at $5$ and at $t$ .

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

Raise $5$ to the power of $3$ .

$lim_{t \to - \infty} - \frac{1}{3} \cdot \frac{1}{125} + \frac{1}{3} t^{- 3}$

Step 4.2.3

Multiply $\frac{1}{125}$ by $\frac{1}{3}$ .

$lim_{t \to - \infty} - \frac{1}{125 \cdot 3} + \frac{1}{3} t^{- 3}$

Step 4.2.4

Multiply $125$ by $3$ .

$lim_{t \to - \infty} - \frac{1}{375} + \frac{1}{3} t^{- 3}$

Step 4.2.5

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

$lim_{t \to - \infty} - \frac{1}{375} + \frac{t^{- 3}}{3}$

Step 4.2.6

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

$lim_{t \to - \infty} - \frac{1}{375} + \frac{1}{3 t^{3}}$

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 $- \infty$ .

$- lim_{t \to - \infty} \frac{1}{375} + lim_{t \to - \infty} \frac{1}{3 t^{3}}$

Step 5.1.2

Evaluate the limit of $\frac{1}{375}$ which is constant as $t$ approaches $- \infty$ .

$- \frac{1}{375} + lim_{t \to - \infty} \frac{1}{3 t^{3}}$

Step 5.1.3

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

$- \frac{1}{375} + \frac{1}{3} lim_{t \to - \infty} \frac{1}{t^{3}}$

Step 5.2

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

$- \frac{1}{375} + \frac{1}{3} \cdot 0$

Step 5.3

Simplify the answer.

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Step 5.3.1

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

$- \frac{1}{375} + 0$

Step 5.3.2

Add $- \frac{1}{375}$ and $0$ .

$- \frac{1}{375}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{375}$

Decimal Form:

$- 0.002\overline{6}$