Calculus Examples

$\int_{0}^{k} \frac{x}{x^{2} + 4} d x = \frac{1}{2} \ln (4)$

Step 1

Let $u = x^{2} + 4$ . 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 1.1

Let $u = x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [4]$

Step 1.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} + 4$ .

$u_{lower} = 0^{2} + 4$

Step 1.3

Simplify.

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Step 1.3.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 + 4$

Step 1.3.2

Add $0$ and $4$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} + 4$ .

$u_{upper} = k^{2} + 4$

Step 1.5

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

$u_{lower} = 4$

$u_{upper} = k^{2} + 4$

Step 1.6

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

$\int_{4}^{k^{2} + 4} \frac{1}{u} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

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

$\int_{4}^{k^{2} + 4} \frac{1}{u \cdot 2} d u$

Step 2.2

Move $2$ to the left of $u$ .

$\int_{4}^{k^{2} + 4} \frac{1}{2 u} d u$

Step 3

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

$\frac{1}{2} \int_{4}^{k^{2} + 4} \frac{1}{u} d u$

Step 4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} \ln (| u |)]_{4}^{k^{2} + 4}$

Step 5

Evaluate $\ln (| u |)$ at $k^{2} + 4$ and at $4$ .

$\frac{1}{2} (\ln (| k^{2} + 4 |) - \ln (| 4 |))$

Step 6

Simplify.

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Step 6.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{2} \ln (\frac{| k^{2} + 4 |}{| 4 |})$

Step 6.2

Combine $\frac{1}{2}$ and $\ln (\frac{| k^{2} + 4 |}{| 4 |})$ .

$\frac{\ln (\frac{| k^{2} + 4 |}{| 4 |})}{2}$

Step 7

Reorder terms.

$\frac{1}{2} \ln (\frac{1}{| 4 |} | k^{2} + 4 |)$

Step 8

Simplify.

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Step 8.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{1}{2} \ln (\frac{1}{4} | k^{2} + 4 |)$

Step 8.2

Combine $\frac{1}{4}$ and $| k^{2} + 4 |$ .

$\frac{1}{2} \ln (\frac{| k^{2} + 4 |}{4})$

Step 8.3

Combine $\frac{1}{2}$ and $\ln (\frac{| k^{2} + 4 |}{4})$ .

$\frac{\ln (\frac{| k^{2} + 4 |}{4})}{2}$