Calculus Examples

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

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

One to any power is one.

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Evaluate the limit.

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Step 5.4.1

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

$0 + 1$

Step 5.4.2

Add $0$ and $1$ .

$1$