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Question: How do you find the indefinite integral of \(\int {\dfrac{1}{{3{x^2} + 3}}dx} \)...

How do you find the indefinite integral of ∫13x2+3dx\int {\dfrac{1}{{3{x^2} + 3}}dx}

Explanation

Solution

According to the question we have to determine the indefinite integral of∫13x2+3dx\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:
⇒∫1x2+1=tan⁡−1x+C.....................(A)\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,
=∫13(x2+1)dx= \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,
=13∫1(x2+1)dx= \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,
⇒13tan⁡−1x+C\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 ⇒13tan⁡−1x+C \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 13\dfrac{1}{3} as a common term we just have to use the formula (A) as mentioned in the solution hint.