Question

this is similar to section 4.2 problem 14:
determine the indefinite integral. use capital c for the free constant.

∫4 ⁶√x dx=

hint: follow example 6.

resources

Explanation:

Step1: Rewrite the integrand

Rewrite \(\sqrt[6]{x}\) as \(x^{\frac{1}{6}}\). Then the integral becomes \(\int4x^{\frac{1}{6}}dx\).

Step2: Use the power rule for integration

The power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(\int4x^{\frac{1}{6}}dx\), since \(\int kf(x)dx=k\int f(x)dx\) (\(k = 4\) is a constant), we have \(4\int x^{\frac{1}{6}}dx\).
Applying the power rule: \(4\times\frac{x^{\frac{1}{6}+1}}{\frac{1}{6}+1}+C\).

Step3: Simplify the expression

\(\frac{1}{6}+1=\frac{1 + 6}{6}=\frac{7}{6}\). Then \(4\times\frac{x^{\frac{7}{6}}}{\frac{7}{6}}+C\).
\(4\times\frac{6}{7}x^{\frac{7}{6}}+C=\frac{24}{7}x^{\frac{7}{6}}+C\).

Answer:

\(\frac{24}{7}x^{\frac{7}{6}}+C\)