  We present the possibilities provided by the MathPartner service of
calculating definite and indefinite integrals. MathPartner contains software
implementation of the Risch algorithm and provides users with the ability to
compute antiderivatives for elementary functions. Certain integrals, including
improper integrals, can be calculated using numerical algorithms. In this case,
every user has the ability to indicate the required accuracy with which he
needs to know the numerical value of the integral. We highlight special
functions allowing us to calculate complete elliptic integrals. These include
functions for calculating the arithmetic-geometric mean and the
geometric-harmonic mean, which allow us to calculate the complete elliptic
integrals of the first kind. The set also includes the modified
arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to
calculate the complete elliptic integrals of the second kind as well as the
circumference of an ellipse. The Lagutinski algorithm is of particular
interest. For given differentiation in the field of bivariate rational
functions, one can decide whether there exists a rational integral. The
algorithm is based on calculating the Lagutinski determinant. Mikhail
Lagutinski (1871--1915) had worked at Kharkiv (Ukraine). This year we are
celebrating his 150th anniversary.
