Question

What is the integral of the quotient?

Answer 1

Hint: To get the answer of this question firstly we should know about the integration and the meaning of the quotient. After knowing this, consider some examples of the quotient and calculate for the value of the integral. After this calculation you are able to conclude for the answer of this question. Now try it by yourself.

Complete step-by-step solution:
Our first step is to understand what the meaning of integration is and what is the use of integration.
The integrаtiоn denоtes the summаtiоn оf disсrete dаtа. The integrаl is саlсulаted tо find the funсtiоns whiсh will describe the аreа, disрlасement, vоlume, thаt оссurs due tо а соlleсtiоn оf smаll dаtа, whiсh саnnоt be meаsured singulаrly. In а brоаd sense, in саlсulus, the ideа оf limit is used where аlgebrа аnd geоmetry аre imрlemented. Limits helр us in the study оf the result оf роints оn а grарh suсh аs hоw they get сlоser tо eасh other until their distance is аlmоst zerо. We knоw thаt there аre twо mаjоr tyрes оf саlсulus . The first one is differential calculus and the other is integral calculus.
We know that the quotient comprises the division of two quantities or values or in other words we can say that the quotient is obtained by dividing the one quantity by the other quantity.
Let us consider that the integral of the quotient is x$${\displaystyle \frac{f(x)}{g(x)}}$$ . Both numerator and denominator are the function of $$x$$.
This integral is represented as
$$\int {\displaystyle \frac{f(x)}{g(x)}}dx$$
So we can say that the integral of any quotient totally depends upon the quotient itself. Like if the value of $${\displaystyle \frac{f(x)}{g(x)}}$$ is $${\displaystyle \frac{1}{x}}$$, then the integral of the $${\displaystyle \frac{f(x)}{g(x)}}$$is represented as
$$\int {\displaystyle \frac{1}{x}}dx$$
The integration of this function is,
$$\Rightarrow \ln (x)$$
And if the value of the $${\displaystyle \frac{f(x)}{g(x)}}$$ is $${\displaystyle \frac{1}{x^2}}$$, then its integral will be represented as
$$\int {\displaystyle \frac{1}{x^2}}dx$$
The integration of this function is,
$$\Rightarrow -{\displaystyle \frac{1}{x}}$$
From this it is clear that the integral of the quotient varies from quotient to quotient.

Note: Integration is the inverse process of differentiation. There are two types of integrals, first one is definite integral and the second one is indefinite integral. Definite integrals are the integral which has an upper and lower limit. But for the indefinite integral there are no upper and lower limits.
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