Calculus Examples

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

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

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

Step 4.2.2

Multiply $- 2$ by $1$ .

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

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 $\infty$ .

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

Step 5.2

Since the function $t^{\frac{1}{2}}$ approaches $\infty$ , the positive constant $2$ times the function also approaches $\infty$ .

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Step 5.2.1

Consider the limit with the constant multiple $2$ removed.

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

Step 5.2.2

Rewrite $t^{\frac{1}{2}}$ as $\sqrt{t}$ .

$lim_{t \to \infty} \sqrt{t} - lim_{t \to \infty} 2$

Step 5.2.3

As $t$ approaches $\infty$ for radicals, the value goes to $\infty$ .

$\infty - lim_{t \to \infty} 2$

Step 5.3

Evaluate the limit.

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Step 5.3.1

Evaluate the limit of $2$ which is constant as $t$ approaches $\infty$ .

$\infty - 1 \cdot 2$

Step 5.3.2

Simplify the answer.

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Step 5.3.2.1

Multiply $- 1$ by $2$ .

$\infty - 2$

Step 5.3.2.2

Infinity plus or minus a number is infinity.

$\infty$