Question

this is similar to section 4.2 problem 20:
determine the indefinite integral. use capital c for the free constant.

hint: follow example 6.

resources
ebook

Explanation:

Step1: Rewrite the integrand

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

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 \(\int 2x^{-4}dx\), since \(\int kf(x)dx=k\int f(x)dx\) (where \(k = 2\) and \(f(x)=x^{-4}\)), we have \(2\int x^{-4}dx\). Using the power rule with \(n=-4\), we get \(2\times\frac{x^{-4 + 1}}{-4+1}+C\).

Step3: Simplify the expression

\(2\times\frac{x^{-3}}{-3}+C=-\frac{2}{3x^{3}}+C\)

Answer:

\(-\frac{2}{3x^{3}}+C\)