Calculus Examples

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

Step 1

Let $u = x + 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $x + 2$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int \frac{u - 2}{\sqrt[4]{u}} d u$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{u}$ as $u^{\frac{1}{4}}$ .

$\int \frac{u - 2}{u^{\frac{1}{4}}} d u$

Step 2.2

Move $u^{\frac{1}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\int (u - 2) {(u^{\frac{1}{4}})}^{- 1} d u$

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{4}})}^{- 1}$ .

Tap for more steps...

Step 2.3.1

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

$\int (u - 2) u^{\frac{1}{4} \cdot - 1} d u$

Step 2.3.2

Combine $\frac{1}{4}$ and $- 1$ .

$\int (u - 2) u^{\frac{- 1}{4}} d u$

Step 2.3.3

Move the negative in front of the fraction.

$\int (u - 2) u^{- \frac{1}{4}} d u$

Step 3

Expand $(u - 2) u^{- \frac{1}{4}}$ .

Tap for more steps...

Step 3.1

Apply the distributive property.

$\int u \cdot u^{- \frac{1}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 3.2

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

$\int u^{1} u^{- \frac{1}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 3.3

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

$\int u^{1 - \frac{1}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 3.4

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

$\int u^{\frac{4}{4} - \frac{1}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 3.5

Combine the numerators over the common denominator.

$\int u^{\frac{4 - 1}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 3.6

Subtract $1$ from $4$ .

$\int u^{\frac{3}{4}} - 2 u^{- \frac{1}{4}} d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{\frac{3}{4}} d u + \int - 2 u^{- \frac{1}{4}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{3}{4}}$ with respect to $u$ is $\frac{4}{7} u^{\frac{7}{4}}$ .

$\frac{4}{7} u^{\frac{7}{4}} + C + \int - 2 u^{- \frac{1}{4}} d u$

Step 6

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$\frac{4}{7} u^{\frac{7}{4}} + C - 2 \int u^{- \frac{1}{4}} d u$

Step 7

By the Power Rule, the integral of $u^{- \frac{1}{4}}$ with respect to $u$ is $\frac{4}{3} u^{\frac{3}{4}}$ .

$\frac{4}{7} u^{\frac{7}{4}} + C - 2 (\frac{4}{3} u^{\frac{3}{4}} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

$\frac{4}{7} u^{\frac{7}{4}} - 2 (\frac{4}{3}) u^{\frac{3}{4}} + C$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Combine $- 2$ and $\frac{4}{3}$ .

$\frac{4}{7} u^{\frac{7}{4}} + \frac{- 2 \cdot 4}{3} u^{\frac{3}{4}} + C$

Step 8.2.2

Multiply $- 2$ by $4$ .

$\frac{4}{7} u^{\frac{7}{4}} + \frac{- 8}{3} u^{\frac{3}{4}} + C$

Step 8.2.3

Move the negative in front of the fraction.

$\frac{4}{7} u^{\frac{7}{4}} - \frac{8}{3} u^{\frac{3}{4}} + C$

Step 9

Replace all occurrences of $u$ with $x + 2$ .

$\frac{4}{7} {(x + 2)}^{\frac{7}{4}} - \frac{8}{3} {(x + 2)}^{\frac{3}{4}} + C$