Calculus Examples

$\int_{100}^{\infty} \frac{x}{\sqrt{1 + x^{2}}} d x$

Step 1

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

$lim_{t \to \infty} \int_{100}^{t} \frac{x}{\sqrt{1 + x^{2}}} d x$

Step 2

Let $u = 1 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 1 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $1 + x^{2}$ .

$\frac{d}{d x} [1 + x^{2}]$

Step 2.1.2

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$

Step 2.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 2 x$

Step 2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.2

Substitute the lower limit in for $x$ in $u = 1 + x^{2}$ .

$u_{lower} = 1 + 100^{2}$

Step 2.3

Simplify.

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Step 2.3.1

Raise $100$ to the power of $2$ .

$u_{lower} = 1 + 10000$

Step 2.3.2

Add $1$ and $10000$ .

$u_{lower} = 10001$

Step 2.4

Substitute the upper limit in for $x$ in $u = 1 + x^{2}$ .

$u_{upper} = 1 + t^{2}$

Step 2.5

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

$u_{lower} = 10001$

$u_{upper} = 1 + t^{2}$

Step 2.6

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

$lim_{t \to \infty} \int_{10001}^{1 + t^{2}} \frac{1}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

$lim_{t \to \infty} \int_{10001}^{1 + t^{2}} \frac{1}{\sqrt{u} \cdot 2} d u$

Step 3.2

Move $2$ to the left of $\sqrt{u}$ .

$lim_{t \to \infty} \int_{10001}^{1 + t^{2}} \frac{1}{2 \sqrt{u}} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} \frac{1}{\sqrt{u}} d u$

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 $u^{\frac{1}{2}}$ .

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} \frac{1}{u^{\frac{1}{2}}} d u$

Step 5.2

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

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} {(u^{\frac{1}{2}})}^{- 1} d u$

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, ${(a^{m})}^{n} = a^{m n}$ .

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} u^{\frac{1}{2} \cdot - 1} d u$

Step 5.3.2

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

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} u^{\frac{- 1}{2}} d u$

Step 5.3.3

Move the negative in front of the fraction.

$lim_{t \to \infty} \frac{1}{2} \int_{10001}^{1 + t^{2}} u^{- \frac{1}{2}} d u$

Step 6

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

$lim_{t \to \infty} \frac{1}{2} 2 u^{\frac{1}{2}}]_{10001}^{1 + t^{2}}$

Step 7

Combine $\frac{1}{2}$ and $2 u^{\frac{1}{2}}]_{10001}^{1 + t^{2}}$ .

$lim_{t \to \infty} \frac{2 u^{\frac{1}{2}}]_{10001}^{1 + t^{2}}}{2}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $2 u^{\frac{1}{2}}$ at $1 + t^{2}$ and at $10001$ .

$lim_{t \to \infty} \frac{(2 {(1 + t^{2})}^{\frac{1}{2}}) - 2 \cdot 10001^{\frac{1}{2}}}{2}$

Step 8.2

Simplify.

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Step 8.2.1

Factor $2$ out of $2 {(1 + t^{2})}^{\frac{1}{2}}$ .

$lim_{t \to \infty} \frac{2 ({(1 + t^{2})}^{\frac{1}{2}}) - 2 \cdot 10001^{\frac{1}{2}}}{2}$

Step 8.2.2

Factor $2$ out of $- 2 \cdot 10001^{\frac{1}{2}}$ .

$lim_{t \to \infty} \frac{2 ({(1 + t^{2})}^{\frac{1}{2}}) + 2 (- 10001^{\frac{1}{2}})}{2}$

Step 8.2.3

Factor $2$ out of $2 ({(1 + t^{2})}^{\frac{1}{2}}) + 2 (- 10001^{\frac{1}{2}})$ .

$lim_{t \to \infty} \frac{2 ({(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}})}{2}$

Step 8.2.4

Cancel the common factors.

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Step 8.2.4.1

Factor $2$ out of $2$ .

$lim_{t \to \infty} \frac{2 ({(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}})}{2 (1)}$

Step 8.2.4.2

Cancel the common factor.

$lim_{t \to \infty} \frac{2 ({(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}})}{2 \cdot 1}$

Step 8.2.4.3

Rewrite the expression.

$lim_{t \to \infty} \frac{{(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}}}{1}$

Step 8.2.4.4

Divide ${(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}}$ by $1$ .

$lim_{t \to \infty} {(1 + t^{2})}^{\frac{1}{2}} - 10001^{\frac{1}{2}}$

Step 9

Evaluate the limit.

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Step 9.1

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

$lim_{t \to \infty} {(1 + t^{2})}^{\frac{1}{2}} - lim_{t \to \infty} 10001^{\frac{1}{2}}$

Step 9.2

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

$lim_{t \to \infty} \sqrt{1 + t^{2}} - lim_{t \to \infty} 10001^{\frac{1}{2}}$

Step 9.3

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

$\infty - lim_{t \to \infty} 10001^{\frac{1}{2}}$

Step 9.4

Evaluate the limit of $10001^{\frac{1}{2}}$ which is constant as $t$ approaches $\infty$ .

$\infty - 10001^{\frac{1}{2}}$

Step 9.5

Infinity plus or minus a number is infinity.

$\infty$