Calculus Examples

$\int \frac{\ln (x)}{x \sqrt{1 + \ln (x)}} d x$

Step 1

Let $u = 1 + \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 + \ln (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 + \ln (x)$ .

$\frac{d}{d x} [1 + \ln (x)]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [\ln (x)]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [\ln (x)]$

Step 1.1.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$0 + \frac{1}{x}$

Step 1.1.4

Add $0$ and $\frac{1}{x}$ .

$\frac{1}{x}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{\ln (e^{u - 1})}{\sqrt{u}} d u$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Simplify.

Tap for more steps...

Step 2.1.1

Use logarithm rules to move $u - 1$ out of the exponent.

$\int \frac{(u - 1) \ln (e)}{\sqrt{u}} d u$

Step 2.1.2

The natural logarithm of $e$ is $1$ .

$\int \frac{(u - 1) \cdot 1}{\sqrt{u}} d u$

Step 2.1.3

Multiply $u - 1$ by $1$ .

$\int \frac{u - 1}{\sqrt{u}} d u$

Step 2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1

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

$\int \frac{u - 1}{u^{\frac{1}{2}}} d u$

Step 2.2.2

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

$\int (u - 1) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 2.2.3

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

Tap for more steps...

Step 2.2.3.1

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

$\int (u - 1) u^{\frac{1}{2} \cdot - 1} d u$

Step 2.2.3.2

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

$\int (u - 1) u^{\frac{- 1}{2}} d u$

Step 2.2.3.3

Move the negative in front of the fraction.

$\int (u - 1) u^{- \frac{1}{2}} d u$

Step 3

Expand $(u - 1) u^{- \frac{1}{2}}$ .

Tap for more steps...

Step 3.1

Apply the distributive property.

$\int u \cdot u^{- \frac{1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 3.2

Raise $u$ to the power of $1$ .

$\int u^{1} u^{- \frac{1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 - \frac{1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 3.4

Write $1$ as a fraction with a common denominator.

$\int u^{\frac{2}{2} - \frac{1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 3.5

Combine the numerators over the common denominator.

$\int u^{\frac{2 - 1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 3.6

Subtract $1$ from $2$ .

$\int u^{\frac{1}{2}} - 1 u^{- \frac{1}{2}} d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{\frac{1}{2}} d u + \int - 1 u^{- \frac{1}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}} + C + \int - 1 u^{- \frac{1}{2}} d u$

Step 6

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{2}{3} u^{\frac{3}{2}} + C - \int u^{- \frac{1}{2}} d u$

Step 7

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

$\frac{2}{3} u^{\frac{3}{2}} + C - (2 u^{\frac{1}{2}} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

$\frac{2}{3} u^{\frac{3}{2}} - 1 \cdot 2 u^{\frac{1}{2}} + C$

Step 8.2

Multiply $- 1$ by $2$ .

$\frac{2}{3} u^{\frac{3}{2}} - 2 u^{\frac{1}{2}} + C$

Step 9

Replace all occurrences of $u$ with $1 + \ln (x)$ .

$\frac{2}{3} {(1 + \ln (x))}^{\frac{3}{2}} - 2 {(1 + \ln (x))}^{\frac{1}{2}} + C$