Question

this is section 4.2 problem 28:
determine the indefinite integral. use capital c for the free constant.
$$ int \frac { - 4 } { sqrt 3 { x ^ { 2 } } } d x = $$
hint: follow example 6.
resources
ebook

Explanation:

Step1: Rewrite the integrand

Rewrite \(\frac{-4}{\sqrt[3]{x^{2}}}\) as \(-4x^{-\frac{2}{3}}\) using the rule \(\frac{1}{\sqrt[n]{x^{m}}}=x^{-\frac{m}{n}}\). So the integral becomes \(\int - 4x^{-\frac{2}{3}}dx\).

Step2: Apply the power rule for integration

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(n=-\frac{2}{3}\), we have:

$$
LATEXBLOCK0
$$

Step3: Simplify the expression

\(-4\times\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+C=-4\times3x^{\frac{1}{3}}+C=-12x^{\frac{1}{3}}+C\)

Answer:

\(-12\sqrt[3]{x}+C\)