Integration for Physics - JEE PDF Download

What is Integration?

• Integration is an inverse process of differentiation.
• It is the process of finding the function whose derivative is given.
• Suppose F(x) is the derivative of the function f(x) w.r.t.x. Then we can write
  
• Now, if we are given the derivative F(x) and we have to find the function f(x) then this can be done with the help of Integral Calculus.
• The process of finding the function whose derivative is given is called integration.

Definition: If the derivative of a function f(x) is F(x) then f(x) is called the integral of F(x) with respect to x. The integration of a function can be written as ∫ F(x)dx = f (x). The function F(x) whose integral is f(x) is called Integrand. 

Fundamental Formulae of Integration

• Integration by parts is a technique used to evaluate integrals of the form  where u and dv are differentiable functions of x. The integration by parts formula is derived from the product rule for differentiation and provides a method for breaking down complicated integrals into simpler ones.
• The formula is given by:

where:

- u is a differentiable function of x,

- dv is a differentiable function of x,

- du is the derivative of u with respect to x,

- v is the antiderivative of dv with respect to x.

• The formula essentially states that the integral of the product of two functions, \(u\) and \(dv\), is equal to the product of u and v minus the integral of the product of v and du.
• The process of integration by parts is often summarized using the mnemonic device "ILATE," which stands for Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, and Exponential. When choosing u and dv, it is often helpful to select u from the functions in this order, as it provides a guideline for choosing u in a way that simplifies the integral.
• The integration by parts formula is particularly useful when dealing with integrals involving products of functions, especially when one factor becomes simpler when differentiated and the other becomes simpler when integrated.
• The formula can be applied iteratively if needed, and multiple applications may be necessary to fully evaluate the integral. Integration by parts is a valuable tool in calculus for solving a variety of integrals and is often employed in the context of finding antiderivatives and definite integrals.

Solved Examples of Integration