Question

this is similar to section 4.2 problem 18: determine the indefinite integral. use capital c for the free constant.

$$\int - 2 \sqrt 5 { x ^ { 4 } } d x =$$

hint: follow example 6. resources ebook

Explanation:

Step1: Rewrite the integrand

Rewrite \(\sqrt[5]{x^{4}}\) as \(x^{\frac{4}{5}}\). So the integral becomes \(\int - 2x^{\frac{4}{5}}dx\).

Step2: Use 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 \(\int - 2x^{\frac{4}{5}}dx\), since \(\int kf(x)dx=k\int f(x)dx\) (\(k=-2\) is a constant), we have \(-2\int x^{\frac{4}{5}}dx\).
Applying the power - rule: \(-2\times\frac{x^{\frac{4}{5}+1}}{\frac{4}{5}+1}+C\).

Step3: Simplify the expression

First, calculate \(\frac{4}{5}+1=\frac{4 + 5}{5}=\frac{9}{5}\).
Then \(-2\times\frac{x^{\frac{9}{5}}}{\frac{9}{5}}+C=-2\times\frac{5}{9}x^{\frac{9}{5}}+C=-\frac{10}{9}x^{\frac{9}{5}}+C\).

Answer:

\(-\frac{10}{9}x^{\frac{9}{5}}+C\)