Methods of Integration Video Lecture | Mathematics for Competitive Exams

Ans. Integration is a mathematical operation that calculates the area under a curve or the accumulation of quantities over a given interval. It is the reverse process of differentiation and helps in finding antiderivatives of functions.

Ans. There are several methods of integration, including: 1. Direct Integration: This method involves finding the antiderivative of a function using basic integration rules. 2. Integration by Parts: This method is used when the integrand is a product of two functions. It involves applying a specific formula derived from the product rule of differentiation. 3. Substitution Method: Also known as the u-substitution method, it involves substituting a new variable to simplify the integrand and then integrating with respect to the new variable. 4. Partial Fractions: This method is used to integrate rational functions by decomposing them into simpler fractions. 5. Trigonometric Substitution: This method is employed when the integrand contains square roots of quadratic expressions. It involves substituting trigonometric functions to simplify the integrand.

Ans. Direct integration involves finding the antiderivative of a function using basic integration rules. To perform direct integration, follow these steps: 1. Identify the function to be integrated. 2. Apply the power rule, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1. 3. If necessary, apply the constant multiple rule, which allows you to bring the constant outside the integral. 4. If necessary, apply the sum rule, which states that the integral of the sum of two functions is equal to the sum of their integrals. 5. Continue simplifying the expression until an antiderivative is obtained.

Ans. The substitution method, also known as the u-substitution method, should be used when the integrand consists of a composition of functions, particularly when a function and its derivative are present. It is most effective when the integrand involves a chain rule or a composite function. By substituting a new variable (u), the integral is transformed into a simpler form, making it easier to solve.

Ans. Deciding which integration method to use depends on the form and complexity of the integrand. Here are some guidelines: 1. For basic functions and polynomials, direct integration is usually straightforward. 2. If the integrand is a product of two functions, integration by parts is suitable. 3. When the integrand contains a square root of a quadratic expression, trigonometric substitution is helpful. 4. If the integrand involves rational functions, partial fractions can be applied. 5. If the integrand consists of a composition of functions or a function and its derivative, the substitution method is often effective. Consider the form of the integrand and choose the method that simplifies the integral or reduces it to a known form.