Question

How do I evaluate \[\int {\dfrac{{3x}}{{{x^2} + 1}}} \,dx\] ?

Answer 1

Hint: Here the question is related to the integration, here they have not mentioned the find the integral. But in the question the integration symbol is used and it implies that we have to integrate the function. By using the standard formulas we find the solution.

Complete step-by-step answer:
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. We know that the integration is inverse of the differentiation.
Now consider the given question
 \[\int {\dfrac{{3x}}{{{x^2} + 1}}} \,dx\]
The function is an algebraic expression, where the algebraic expression is a combination of variables and the constants.
Now we divide and multiply by 2 to the given inequality
 \[ \Rightarrow \int {\dfrac{{3x}}{{{x^2} + 1}}} \,\dfrac{2}{2}dx\]
Let we take 3 in the numerator and 2 in the denominator from the integral sign
 \[ \Rightarrow \dfrac{3}{2}\int {\dfrac{{2x}}{{{x^2} + 1}}} \,dx\]
the above term is in the form of \[\int {\dfrac{{f'(x)}}{{f(x)}}} \,dx\] and we have the standard formula for this form and it is given as \[\int {\dfrac{{f'(x)}}{{f(x)}}} \,dx = \ln |f(x)| + c\] ,c is the integrating constant . so applying the formula for the above equation we get
 \[ \Rightarrow \dfrac{3}{2}\ln |{x^2} + 1| + c\]
The c is the integrating constant.
Hence we have evaluated the given question and hence we have determined the solution for the given question
So, the correct answer is “ \[ \Rightarrow \dfrac{3}{2}\ln |{x^2} + 1| + c\]”.

Note: In integration we have standard integration formulas for some of the functions. We have to examine the given function properly and then we have to determine by which way we have to simplify it. we must to know the formula \[\int {\dfrac{{f'(x)}}{{f(x)}}} \,dx = \ln |f(x)| + c\] , by using this we simplify the function and hence obtain the answer.