This tutorial provides a detailed solution of problems of integral calculus and differential equations integration. Practical training prior to basic theoretical information, reference data and formulas. The book is intended for students of higher technical educational institutions. Practical tasks are distributed on the following topics:
1. primitive function and indefinite integral, properties of indefinite integral. Direct integration.
2. Integration of exponential and trigonometric functions.
3. Replacing a variable in the indefinite integral (substitution method).
Integration by parts.
4. simple fractions. The decomposition of a rational fraction into partial fractions. Integration of rational fractions.
5. Integration of expressions containing trigonometric functions. Integration of algebraic irrationalities.
6. The integral sum. Definite integral and its basic properties. Problems of mechanics and physics, leading to the calculation of the limit of Riemann sums. The calculation of the definite integral as the limit of Riemann sums.
7. Replacement of a variable in a definite integral. Integration by parts. Mean value theorem.
8.Nesobstvennye integrals over infinite and the integral of discontinuous functions. The principle of comparison of improper integrals with positive integrands.
9. Approximate calculation of integrals using rectangles, trapezoids and Simpson (parabolas formula).
10. Applications of the definite integral in geometry. Determining the areas of plane figures.
11. Applications of the definite integral geometry (continued): the arc length of the plane curve, the amount of rotation of the body, the surface of a body of revolution.
12. The differential equations of first order.
13. Differential equations of higher order, permitting reduction of order.
14. Linear differential equations of higher orders.
15. Linear inhomogeneous differential equation.
16. Euler equation. Systems of linear differential equations with constant coefficients (basic concepts).