Question

What is the integral of \[\int {\dfrac{1}{{{x^2} + 1}}dx} \] ?

Answer 1

Hint: Here we are asked to find the integral of the given expression. For integration, we will use the standard formula that is available in integration. If the expression in the given integral is complex then we can also simplify them before integrating it.
Formula: Formula that we need to know:
 $\int {\dfrac{1}{{{x^2} + 1}}dx = {{\tan }^{-1}}x} $

Complete step by step answer:
 It is given that \[\int {\dfrac{1}{{{x^2} + 1}}dx} \] we aim to integrate this integral.
Before we start integrating it, we must look into whether it can be simplified or not. As we can see that the expression is a fraction whose numerator is one and the denominator contains an algebraic expression with the unknown variable x whose degree is two. So, we cannot do any further simplification in that so we will keep the integral as it is.
Now let us see whether we can use any standard formula to integrate it. From the inverse function, we have $\int {\dfrac{1}{{{x^2} + 1}}dx = {{\tan }^{-1}}x} $ let us use this formula to integrate the given integral.
Consider the given integral \[\int {\dfrac{1}{{{x^2} + 1}}dx} \] from the formula we have that $\int {\dfrac{1}{{{x^2} + 1}}dx = {{\tan }^{-1}}x} $ so on integrating the given integral concerning the variable x we get
$\int {\dfrac{1}{{{x^2} + 1}}dx = {{\tan }^{-1}}x} $
Let us add the integration constant to it.
$\int {\dfrac{1}{{{x^2} + 1}}dx = {{\tan }^{-1}}x} + C$
Thus, we have got the required solution.

Note:
In mathematics, we have two types of integral; they are definite and indefinite integrals. Students should note that indefinite integrals are the ones that do not have any limits (upper and lower limits) so we should not forget to use constant and the definite integrals are the ones that have upper and lower limits.