Calculus Examples

$\int x^{2} \sqrt[4]{4 x - 3} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sqrt[4]{4 x - 3}$ .

$x^{2} (\frac{1}{5} {(4 x - 3)}^{\frac{5}{4}}) - \int \frac{1}{5} {(4 x - 3)}^{\frac{5}{4}} (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{5}$ and ${(4 x - 3)}^{\frac{5}{4}}$ .

$x^{2} \frac{{(4 x - 3)}^{\frac{5}{4}}}{5} - \int \frac{1}{5} {(4 x - 3)}^{\frac{5}{4}} (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{{(4 x - 3)}^{\frac{5}{4}}}{5}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \int \frac{1}{5} {(4 x - 3)}^{\frac{5}{4}} (2 x) d x$

Step 3

Since $\frac{1}{5} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{5} \cdot 2$ out of the integral.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - (\frac{1}{5} \cdot 2 \int {(4 x - 3)}^{\frac{5}{4}} (x) d x)$

Step 4

Combine $\frac{1}{5}$ and $2$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int {(4 x - 3)}^{\frac{5}{4}} (x) d x$

Step 5

Let $u = 4 x - 3$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 4 x - 3$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $4 x - 3$ .

$\frac{d}{d x} [4 x - 3]$

Step 5.1.2

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$

Step 5.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 5.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 5.1.3.2

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

$4 \cdot 1 + \frac{d}{d x} [- 3]$

Step 5.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 3]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$4 + 0$

Step 5.1.4.2

Add $4$ and $0$ .

$4$

Step 5.2

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

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int u^{\frac{5}{4}} (\frac{u}{4} + \frac{3}{4}) \frac{1}{4} d u$

Step 6

Combine $\frac{1}{4}$ and $u^{\frac{5}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4}}}{4} (\frac{u}{4} + \frac{3}{4}) d u$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4}}}{4} \cdot \frac{u}{4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.2

Multiply $\frac{u^{\frac{5}{4}}}{4}$ by $\frac{u}{4}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4}} u}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.3

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

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4}} u^{1}}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.4

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

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4} + 1}}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.5

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

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5}{4} + \frac{4}{4}}}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.6

Combine the numerators over the common denominator.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{5 + 4}{4}}}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.7

Add $5$ and $4$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{9}{4}}}{4 \cdot 4} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.8

Multiply $4$ by $4$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{9}{4}}}{16} + \frac{u^{\frac{5}{4}}}{4} \cdot \frac{3}{4} d u$

Step 7.9

Multiply $\frac{u^{\frac{5}{4}}}{4}$ by $\frac{3}{4}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{9}{4}}}{16} + \frac{u^{\frac{5}{4}} \cdot 3}{4 \cdot 4} d u$

Step 7.10

Multiply $4$ by $4$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{9}{4}}}{16} + \frac{u^{\frac{5}{4}} \cdot 3}{16} d u$

Step 8

Move $3$ to the left of $u^{\frac{5}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} \int \frac{u^{\frac{9}{4}}}{16} + \frac{3 u^{\frac{5}{4}}}{16} d u$

Step 9

Split the single integral into multiple integrals.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\int \frac{u^{\frac{9}{4}}}{16} d u + \int \frac{3 u^{\frac{5}{4}}}{16} d u)$

Step 10

Since $\frac{1}{16}$ is constant with respect to $u$ , move $\frac{1}{16}$ out of the integral.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} \int u^{\frac{9}{4}} d u + \int \frac{3 u^{\frac{5}{4}}}{16} d u)$

Step 11

By the Power Rule, the integral of $u^{\frac{9}{4}}$ with respect to $u$ is $\frac{4}{13} u^{\frac{13}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} (\frac{4}{13} u^{\frac{13}{4}} + C) + \int \frac{3 u^{\frac{5}{4}}}{16} d u)$

Step 12

Combine $\frac{4}{13}$ and $u^{\frac{13}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} (\frac{4 u^{\frac{13}{4}}}{13} + C) + \int \frac{3 u^{\frac{5}{4}}}{16} d u)$

Step 13

Since $\frac{3}{16}$ is constant with respect to $u$ , move $\frac{3}{16}$ out of the integral.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} (\frac{4 u^{\frac{13}{4}}}{13} + C) + \frac{3}{16} \int u^{\frac{5}{4}} d u)$

Step 14

By the Power Rule, the integral of $u^{\frac{5}{4}}$ with respect to $u$ is $\frac{4}{9} u^{\frac{9}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} (\frac{4 u^{\frac{13}{4}}}{13} + C) + \frac{3}{16} (\frac{4}{9} u^{\frac{9}{4}} + C))$

Step 15

Simplify.

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Step 15.1

Combine $\frac{4}{9}$ and $u^{\frac{9}{4}}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{1}{16} (\frac{4 u^{\frac{13}{4}}}{13} + C) + \frac{3}{16} (\frac{4 u^{\frac{9}{4}}}{9} + C))$

Step 15.2

Simplify.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12}) + C$

Step 15.3

Simplify.

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Step 15.3.1

To write $- \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} - \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12}) \cdot \frac{5}{5} + C$

Step 15.3.2

Combine $- \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})$ and $\frac{5}{5}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}}}{5} + \frac{- \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12}) \cdot 5}{5} + C$

Step 15.3.3

Combine the numerators over the common denominator.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} - \frac{2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12}) \cdot 5}{5} + C$

Step 15.3.4

Multiply $5$ by $- 1$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} - 5 (\frac{2}{5}) (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.5

Combine $- 5$ and $\frac{2}{5}$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{- 5 \cdot 2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.6

Multiply $- 5$ by $2$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{- 10}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.7

Cancel the common factor of $- 10$ and $5$ .

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Step 15.3.7.1

Factor $5$ out of $- 10$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{5 \cdot - 2}{5} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.7.2

Cancel the common factors.

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Step 15.3.7.2.1

Factor $5$ out of $5$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{5 \cdot - 2}{5 (1)} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.7.2.2

Cancel the common factor.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{5 \cdot - 2}{5 \cdot 1} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.7.2.3

Rewrite the expression.

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} + \frac{- 2}{1} (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 15.3.7.2.4

Divide $- 2$ by $1$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} - 2 (\frac{u^{\frac{13}{4}}}{52} + \frac{u^{\frac{9}{4}}}{12})}{5} + C$

Step 16

Replace all occurrences of $u$ with $4 x - 3$ .

$\frac{x^{2} {(4 x - 3)}^{\frac{5}{4}} - 2 (\frac{{(4 x - 3)}^{\frac{13}{4}}}{52} + \frac{{(4 x - 3)}^{\frac{9}{4}}}{12})}{5} + C$

Step 17

Reorder terms.

$\frac{1}{5} (x^{2} {(4 x - 3)}^{\frac{5}{4}} - 2 (\frac{1}{52} {(4 x - 3)}^{\frac{13}{4}} + \frac{1}{12} {(4 x - 3)}^{\frac{9}{4}})) + C$