This learning solution provides a clear exposition of essential tools of Indefinite Integration, Definite Integration and Its Applications, Differential Equations.
Integration regarded as reverse of differentiation – Introduction. Indefinite integral and properties of integrals - Indefinite integral and properties of integrals - Example problems. Finding integral of simple function - Example problems. Evaluation of arbitrary constant and determine particular integrals – Example problems.s Various Methods of Integration - Introduction to method of integration - Integration by substitution (Type 1) - Example problems. Integration by substitution (Type II and III) - Integration by substitution (Type II) – Example problem - Integration by substitution (Type III) – Example problems. Integration by substitution (Type IV) - Integrate simple function by substitution (Type IV) – Example problems. Integration by Substitution (Type V and VI) - Integration By Substitution (Type V and VI) – Example problems - Integration by Substitution(Type VI) – Example problems. Integrals products of powers of sinx and cosx - Integrals products of powers of sinx and cosx - Example problems - Integrals products of powers of tanx and secx - Example problems. Integration by nine standard integral (Type I) - Integration by nine standard integral (Type I) – Example problems. Integration by nine standard integral (Type II) - Integration by nine standard integral (Type II) - Example problems. Integration by nine standard integral (Type III) - Integration by nine standard integral (Type III) – Example problems. Integration by nine standard integral (Type IV) - Integration by nine standard integral (Type IV) – Example problems. Integration by nine standard integral (Type V) - Introduction - Example problems. Integration by decomposition or method of partial fractions - Integration by decomposition or method of partial fractions – Example problems - Partial fractions by substitution – Example problems. Integration by parts - Integration by parts – Example problem. Bernoulli’s rule – Bernoulli’s rule - Example programs.
Definite Integral- Introduction to Definite integral – Example problems. Integral calculus - The fundamental theorem of integral calculus - Example problems. Evaluate a definite integral as limit of a sum - Definite Integral as the limit of a sum - Example problems. Properties of definite Integral - Properties of definite integrals – Example problems. Evaluation of definite integral –Example problems. Quadrature – Example problems. Volumes of solids of revolution – Volumes of solids of revolution - Volume of solid of revolution about the x-axis – Example problems. Mean and R.M.S values – Mean and R.M.S values – Example problems. Numerical Integration – Numerical Integration(Introduction) - Numerical Integration - Trapezoidal rule – Example problems – Simpsons 1/3 rule – Example problems. Laplace transform – Laplace Transform of Some Standard Functions – Example problems. First, Second Shifting Theorem and Change of Scale Property - First Shifting Theorem- Example problems - Second Shifting Theorem – Example problems - Change of Scale Property. Laplace Transform of Multiplication by 't' and Division by 't'- Laplace Transform of Multiplication by 't' – Example problems - Laplace Transform of Division by 't' – Example problems. Laplace Transform of Derivatives and Integrals – Laplace Transform of Derivatives - Example problems - Laplace Transform of Integrals – Example problems. Laplace Transform of Periodic Functions - Laplace Transform of Periodic Functions – Example problems. Laplace Transform of Unit Step and Unit Impulse Functions – Laplace Transform of Unit Step Functions - Laplace Transform of Unit Impulse Functions - Example problems. Inverse Laplace Transform – Inverse Laplace Transform – Example problems. Inverse Laplace Transform – Method of Partial Fraction - Example problems - Real time problem. First Shifting Theorem – First Shifting Theorem – Example problems - Second Shifting Theorem - Example problems - Change of Scale of Property – Example problems. Inverse Laplace Transform of Derivatives – Inverse Laplace Transform of Derivatives - Example problems - Inverse Laplace Transform of Integrals – Example problems. Multiplication by Power of s - Multiplication by Power of s – Example problems - Division by s – Example problems. Convolution theorem - Convolution theorem – Example problems. Fourier series - History of Fourier Series - Periodic Function – example problem - Introduction of Fourier Series - Dirichlet Conditions - Determination of Fourier Coefficients – Euler’s Formula – Example problems. Function having points of discontinuity - Function having points of discontinuity – Example problems. Even and odd functions - Fourier Series For Even And Odd Function – Example problems. Half range Fourier series - Introduction to Half range Fourier Series – Example problems. Change of interval - Fourier series for functions having period 2l - Fourier series for even and odd function in (-l, l) – Example problems.
Order and Degree of Differential equation - Differential equation - Order and Degree of Differential equation – Example problems. Formation of a Differential equation - Formation of a Differential equation – Example problems. Solution of a Differential equation - Solution of a Differential equation - Differential equations of first order and first degree. Variable Separable Method - Variable Separable – Example problems. Homogeneous Differential equation - Homogeneous function - Homogeneous Differential equation - Method of solving a Homogeneous equation – Example problems. Example of Homogeneous equations – Example problems. Exact Differential equation - Exact Differential equation – Example problems. Linear differential equation – Linear differential equation – Linear differential equation of first order – Working Rule for solving linear differential equation - Example problems. Bernoulli's Equation - Non-linear differential equation – Example problem - Working rule to solve Bernoulli’s equation – Example equation. Homogeneous linear differential equations of second order with constant coefficients - Homogeneous linear differential equations of second order with constant coefficients - Auxiliary equation (A.E) - Working rule – Example problems. Linear differential equations of higher order - Linear differential equations of higher order - Operator D and Auxiliary equation General solution of f(D)y = 0(Homogeneous linear equation) - Example problems. Find the solution of f(D)y = eax – General solution of f(D)y = X (Non-homogeneous Linear equation) - To Find P.I. when X is of the from X = eax - Example problems. Find the solution of f(D)y = cosax or sinax - Particular Integral of f(D) = X when X = cosax or sinax, where a is any constant - Working rule to evaluate Particular Integral – Example problems. Find the solution of f(D)y = xm – Particular Integral of f(D)y = X when X = xm , m being a positive integer - Example problems.
