Calculus Examples

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

Step 1

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

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

Step 2

Let $u = π x$ . Then $d u = π d x$ , so $\frac{1}{π} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = π x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 2.1.2

Since $π$ is constant with respect to $x$ , the derivative of $π x$ with respect to $x$ is $π \frac{d}{d x} [x]$ .

$π \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$π \cdot 1$

Step 2.1.4

Multiply $π$ by $1$ .

$π$

Step 2.2

Substitute the lower limit in for $x$ in $u = π x$ .

$u_{lower} = π \cdot 1$

Step 2.3

Multiply $π$ by $1$ .

$u_{lower} = π$

Step 2.4

Substitute the upper limit in for $x$ in $u = π x$ .

$u_{upper} = π t$

Step 2.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = π$

$u_{upper} = π t$

Step 2.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$lim_{t \to \infty} \int_{π}^{π t} \frac{1}{\sqrt{u}} \cdot \frac{1}{π} d u$

Step 3

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{π}$ .

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

Step 4

Since $\frac{1}{π}$ is constant with respect to $u$ , move $\frac{1}{π}$ out of the integral.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Combine $\frac{1}{π}$ and $2 u^{\frac{1}{2}}]_{π}^{π t}$ .

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

Step 8

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

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

Step 9

Evaluate the limit.

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Step 9.1

Evaluate the limit.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.3

Evaluate the limit.

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Step 9.3.1

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

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

Step 9.3.2

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

$\frac{\infty - 2 π^{\frac{1}{2}}}{π}$

Step 9.3.3

Simplify the answer.

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Step 9.3.3.1

Infinity plus or minus a number is infinity.

$\frac{\infty}{π}$

Step 9.3.3.2

Infinity divided by anything that is finite and non-zero is infinity.

$\infty$