Question

What happens when you apply the power rule for integration to the function $f\left( x \right) = \dfrac{1}{x}$ ?

Answer 1

Hint: Here we are going to see what happens when applying the integration power rule to the given function. Before that, we will understand the power rule for integration. We also treat each of the special cases such as negative and fractional exponents to integrate functions involving roots and reciprocal powers $x$.

Complete step-by-step answer:
Power rule for integration:
The power rule for integration provides us with a formula that allows us to integrate any function that can be written as a power of $x$. The power rule for integration is an essential step in learning integration.
$ \Rightarrow \int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C} $ .
Given function looking like $f\left( x \right) = \dfrac{1}{x}$can be written using a negative exponent,
$ \Rightarrow f\left( x \right) = {x^{ - 1}}$
Now we can use the power rule, we get,
$ \Rightarrow \int {{x^{ - 1}}dx = \dfrac{{{x^{ - 1 + 1}}}}{{ - 1 + 1}} + C} $
$ \Rightarrow \int {{x^{ - 1}}dx = \dfrac{{{x^0}}}{0} + C} $
\[ \Rightarrow \int {{x^{ - 1}}dx = \dfrac{1}{0} + C} \]
The power rule does not work for this function, because $\dfrac{1}{0}$ is undefined.
So instead of this, we can use the standard result, that is
$\int {\dfrac{1}{x}dx = \log x + C} $ .

Note: The power rule for integrals allows us to find the indefinite (and later the definite) integrals of a variety of functions like polynomials, functions involving roots, and even some rational function; you probably can apply the power rule. As you have seen, the power rule can be used to find simple integrals. The general strategy is always the same if you don’t already have exponents, see if you can write the using exponents. Then, apply the power rule and simplify. Remember you can always check your integration by differentiating and you find the result. If it is correct then your integration also corrects.