METHODS FOR SOLVING DEFINITE INTEGRALS

Аннотация

This article unclear integrals solution two main to the method , that is in pieces integration (partial integration )and variable exchange to themethods dedicated .Thismethods , unclear integrals in calculation complicated functions simplification and tocalculate facilitate for wide is applied .The article firstly breaks down of integration main principles and his/her unclear integrals in solution how application seeing We will go out . This The method usually involves integrals simplertoexpressions separation through thesolution tofind help gives . Key words:Indefiniteintegral, piecewise integration ,variable exchange ,mathematics analysis , integrals solve,functions simplification ,variable exchange In our country, mathematics has been identified as one of the priority areas for the development of science in 2020. Over the past period, a number of systematic works have been carried out aimed at bringing mathematical science and education toa new qualitativestage. Resolution of the President of the Republic of Uzbekistan dated May 7, 2020 No. PQ-4708 “On measures to improve the quality of education and develop scientific research in the field of mathematics”, dated July 9, 2020 “On state support for the further development of mathematical education and sciences, as well as the V.I. In connection with the rapid development of modern branches of mathematics, science and technology, the foundation of which was laid by our great ancestors such as Muhammad Al-Khwarizmi, Ahmad Al-Farghani, Abu Rayhon Beruni, Mirzo Ulugbek, and the resolutions of the State Duma No. PQ-4387 “On measures to radically improve the activities of the Romanovsky Institute of Mathematics”, the in-depth teaching of integrals, one of the main topics in higher education, is currently one of the important issues. This article serves to address theabove-mentioned resolutions and theissues set before us. Taking the above into account, we will study several methods for finding definite integrals of the followingform. If the equality holds for , the function is called the initial function ofa continuous function onthe interval. Given function elementary function to find integration The function is called of integration one how many methods there is is , this in the article in pieces integration method and variables replacement method about stopped Let's go . 1.Integration byparts method . If u(x) and v(x) are differentiable functions if so , in pieces integration for Formula (1) is used. This method is used to find integrals of the form , , , , page828 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue03,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai , , ,. This method in use under the given integral f(x)dx expression he *dv in appearance choice must be , as a result of (1) – right on the side voodoo expression in integration initial integral to find relatively simpleLet itbe . In general he and dv what to choose separately attention to give necessary.If given integral under expression plural and trigonometric or indicative functions from the multiplication consists ofif,then sheis function as polytheism to take to thegoal appropriate willbe . If the integral is function logarithmic and reverse trigonometric of functions in the plural consists of if ,then sheis function as from differentiation thensoda to look​ coming function iftaken good result gives . In pieces integration repetitive in use initially she is with how function what designated if we are then he /she with this function designation necessary , otherwise integral to zero without equal will be. Example1. Findtheintegral. Solution : u=3x+5 and We get uv = cos5x . Then du=3dx, Substituting this expression into formula (1), we obtain the following: Example2. Find theintegral. Solution : (*) Harvest made (*) PCB right by To find the integral, we again use the formula for integration byparts: 2 2 2 2 u = ln3x dv = xdx 2 x x x x xln3xdx = 1 x = ln3x− ln3xdx == ln3x− +C du = dx v = 2 2 2 4 Thisis related to (*) given ,givenfunction integral x 2 page829 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue03,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai w edetermine that. 2. Substitution of variables method . Hypothesis let's do it Let the substitution be performed in the integral . If a continuous function with an inverse has a derivative ,then (2) The formula is valid. will be . (2) to unclear in the integral variable replacement formula It is called . Someincases Getting areplacement will yield good results. The variable replacement given internal simple to look if it brings it , application to the goal appropriatewill be. Uncertain in the integral variable replacement as a result found elementary in function new variable instead of initial variable through expression leaving given of the in