Find The Antiderivative of more complex expressions. Need help finding the antiderivative of this one in particular
$\int_{}^{}$$\frac{dx}{3x²+1}$
Would you kindly also give some tips on how to find the antiderivative of any expressions. I am having problems switching from differential calculus to integral calculus as finding the antiderivative  is not as straight forward as finding the differential of an expression.
 A: SIMPLY: By Formula
You need to know the formula$$\int\frac1{x^2+a^2}dx=\frac1a\arctan\left(\frac xa\right)+c$$So your integral becomes:
$$\begin{align}\int \frac1{3x^2+1}dx&=\frac13\int\frac1{x^2+\frac13}dx=\frac13\int\frac1{x^2+\left(\frac1{\sqrt3}\right)^2}dx\\&=\frac13\times\frac1{1/\sqrt3}\arctan\left(\frac x{1/\sqrt3}\right)+c\\&=\frac1{\sqrt3}\arctan\sqrt3x+c\end{align}$$
ANOTHER METHOD: By Substitution
Let $x=\displaystyle\frac1{\sqrt3}\tan \theta$. So $dx=\displaystyle\frac1{\sqrt3}\sec^2\theta\ d\theta$. Also 
$$3x^2+1=3\left(\frac13\tan^2\theta\right)+1=\tan^2\theta+1=\sec^2\theta$$
Hence your integral becomes
$$\begin{align}
\int\frac{1}{3x^2+1}dx&=\int\frac1{\sec^2\theta}\left(\frac1{\sqrt3}\sec^2\theta\ d\theta\right)\\
&=\int\frac1{\sqrt3}d\theta\\
&=\frac1{\sqrt3}\theta +c\ \ \ \ \ \ \ \ \ \ \cdots(i)
\end{align}$$
Now since $x=\displaystyle\frac1{\sqrt3}\tan\theta$, so $\theta=\arctan\sqrt3x$. Putting this in one we have the required answer as
$$\int\frac1{3x^2+1}dx=\frac{1}{\sqrt3}\arctan\sqrt3x+c$$
A: Not all elementary functions have an elementary antiderivative: there are nonelementary integrals.
Here are some usual integration rules:
https://en.wikipedia.org/wiki/Integration_by_parts
https://en.wikipedia.org/wiki/Integration_by_substitution
https://en.wikipedia.org/wiki/Partial_fraction_decomposition#Application_to_symbolic_integration
https://en.wikipedia.org/wiki/Integration_by_reduction_formulae
https://en.wikipedia.org/wiki/Trigonometric_substitution
https://en.wikipedia.org/wiki/Integral_of_inverse_functions
Here are techniques for integration by parts:
Alcantara, E.: Integrals of composite functions through tabular integration by parts. Asia Pacific Higher Educ. Res. J. 2 (2015) (1)
Alcantara, E.: On the Derivation of Some Reduction Formula through Tabular Integration by Parts. Asia Pacific J. Multidisciplinary Res 3 (2015) (1) 80-84
Mardeli Jandja, M.; Lutfi, M.: The Five Columns Rule in Solving Definite Integration by Parts Through Transformation of Integral Limits. J. Phys.: Conf. Ser. 1028 (2018) 012109
Here are some integration rules that may be known not to everyone:
Will, J.: Produktregel, Quotientenregel, Reziprokenregel, Kettenregel und Umkehrregel für die Integration. 2017
Will, J.: Product rule, quotient rule, reciprocal rule, chain rule and inverse rule for integration. May 2017
Maybe these integration rules are not taught everywhere. They are deduced from the usual general integration rules above.
Here is a hierarchy for applying integration rules and integration methods:
Swartz, J.: Symbolic integration using CLIPS. Dr. Dobb's
A: Generalize this. Start with the integral$$I=\int\frac {\mathrm dx}{a^2+(bx)^2}$$Let $z=\tfrac {bx}a$ so that $\mathrm dz=\tfrac ba\,\mathrm dx$. Thus$$\begin{align*}I & =\frac ab\int\frac {\mathrm dz}{a^2+a^2z^2}\\ & =\frac 1{ab}\int\frac {\mathrm dz}{1+z^2}\end{align*}$$The integral is now easily seen to be $\arctan z+C$. Thus$$\int\frac {\mathrm dx}{a^2+(bx)^2}\color{blue}{=\frac 1{ab}\arctan\left(\frac {bx}a\right)+C}$$Now set $a=1$ and $b=\sqrt3$.
