Step by step solution to "Evaluate the given indefinite integral. ∫(4x3−4x23)dx"

High School
Other

kolonelyf4

Answered question

2021-11-19

Evaluate the given indefinite integral.
$\int (\frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}) d x$

Answer & Explanation

Florence Evans

Beginner2021-11-20Added 16 answers

Step 1
The given integral is
$\int (\frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}) d x$
Step 2
Evaluate the given integral as shown below:
$\int (\frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}) d x = 4 \int (\frac{1}{\sqrt[3]{x}}) d x - 4 \int (\sqrt[3]{x^{2}}) d$
$= 4 \int (x^{- \frac{1}{3}}) d x - 4 \int (x^{\frac{2}{3}}) d x$
$= 4 (\frac{3 x^{\frac{2}{3}}}{2}) - 4 (\frac{3 x^{\frac{5}{3}}}{5})$
$= (6 x^{\frac{2}{3}}) - (\frac{12}{5} x^{\frac{5}{3}}) + C$

Sarythe

Beginner2021-11-21Added 11 answers

Step 1: Use this rule: ${(x^{a})}^{b} = x^{a b}$ .
$\int \frac{4}{\sqrt[3]{x}} - 4 x^{\frac{2}{3}} d x$
Step 2: Use Sum Rule: $\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x$
$\int \frac{4}{\sqrt[3]{x}} d x + \int - 4 x^{\frac{2}{3}} d x$
Step 3: Use Constant Factor Rule: $\int c f (x) d x = c \int f (x) d x$ .
$4 \int \frac{1}{\sqrt[3]{x}} d x + \int - 4 x^{\frac{2}{3}} d x$
Step 4: Use Power Rule: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ .
$\frac{6}{x^{- \frac{2}{3}}} + \int - 4 x^{\frac{2}{3}} d x$
Step 5: Use Power Rule: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ .
$\frac{6}{x^{- \frac{2}{3}}} - \frac{12 x^{\frac{5}{3}}}{5}$
Step 6: Add constant.
$\frac{6}{x^{- \frac{2}{3}}} - \frac{12 x^{\frac{5}{3}}}{5} + C$

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