  This paper presents a novel algorithm to obtain the closed-form
anti-derivative of a function using Deep Neural Network architecture. In the
past, mathematicians have developed several numerical techniques to approximate
the values of definite integrals, but primitives or indefinite integrals are
often non-elementary. Anti-derivatives are necessarily required when there are
several parameters in an integrand and the integral obtained is a function of
those parameters. There is no theoretical method that can do this for any given
function. Some existing ways to get around this are primarily based on either
curve fitting or infinite series approximation of the integrand, which is then
integrated theoretically. Curve fitting approximations are inaccurate for
highly non-linear functions and require a different approach for every problem.
On the other hand, the infinite series approach does not give a closed-form
solution, and their truncated forms are often inaccurate. We claim that using a
single method for all integrals, our algorithm can approximate anti-derivatives
to any required accuracy. We have used this algorithm to obtain the
anti-derivatives of several functions, including non-elementary and oscillatory
integrals. This paper also shows the applications of our method to get the
closed-form expressions of elliptic integrals, Fermi-Dirac integrals, and
cumulative distribution functions and decrease the computation time of the
Galerkin method for differential equations.
