Question

How do you find the indefinite integral of $\int {\dfrac{1}{{3{x^2} + 3}}dx} $

Answer 1

Hint:
According to the question we have to determine the indefinite integral of$\int {\dfrac{1}{{3{x^2} + 3}}dx} $. So, to determine the indefinite integral of the given function first of all we have to rearrange the terms of the given integral which can be done by taking 3 as a common term form the denominator of the integral.
Now, we have to take 3 as a constant term of the integration so that we can determine the required integration of the terms.
Now, we have to integrate the integral after taking 2 as a constant term with the help of the formula which is as mentioned below:

Formula used:
$ \Rightarrow \int {\dfrac{1}{{{x^2} + 1}} = {{\tan }^{ - 1}}x} + C.....................(A)$
Where, C is the constant term.

Complete step by step solution:
Step 1: function first of all we have to rearrange the terms of the given integral which can be done by taking 3 as a common term form the denominator of the integral. Hence,
$ = \int {\dfrac{1}{{3({x^2} + 1)}}dx} $
Step 2: Now, we have to take 3 as a constant term of the integration so that we can determine the required integration of the terms. Hence,
$ = \dfrac{1}{3}\int {\dfrac{1}{{({x^2} + 1)}}dx} $
Step 3: Now, we have to integrate the integral after taking 2 as a constant term with the help of the formula (A) which is as mentioned in the solution hint. Hence,
$ \Rightarrow \dfrac{1}{3}{\tan ^{ - 1}}x + C$

Hence, with the help of the formula (A) which is as mentioned in the solution hint we have determined the required integral which is $ \Rightarrow \dfrac{1}{3}{\tan ^{ - 1}}x + C$.

Note:
1) To determine the required integral it is necessary that we have to take the integer 3 as a common term and then we have to make it as a common term.
2) To determine the integration of the function after taking $\dfrac{1}{3}$ as a common term we just have to use the formula (A) as mentioned in the solution hint.