Integration by Parts

∫f(x) g(x) dx = f(x)∫g(x)dx – ∫[f'(x)∫g(x)dx]dx

This is the basic formula which is used to integrate products of two functions by parts.

If we consider f as the first function and g as the second function, then this formula may be pronounced as:
“The integral of the product of two functions = (first function) × (integral of the second function) – Integral of [(differential coefficient of the first function) × (integral of the second function)]”.

ILATE Rule

Identify the function that comes first on the following list and select it as f(x).

ILATE stands for:

I: Inverse trigonometric functions : tan-1 x, sec-1 x, sin-1 x etc.

L: Logarithmic functions : ln x, log5(x), etc.

A: Algebraic functions. such as,x ,x2 etc..

T: Trigonometric functions, such as sin x, cos x, tan x etc.

E: Exponential functions.

Example : Find ∫ x cos x dx

Solution: Let, The first function = f(x) = x and the second function = g(x) = cos x. Therefore, according to integration by parts, we have

∫ x cos x dx = x ∫ cos x dx – ∫ [(dx/dt) ∫ cos x dx] dx = x sin x – ∫ sin x dx
= x sin x + cos x + C.

Let’s try the other way round. Let, the first function = f(x) = cos x and the second function = g(x) = x. Therefore,
∫ x cos x dx = cos x ∫ x dx – ∫ {[d(cos x)/ dx] ∫ x } dx
= (cos x) (x2/2) + ∫ (sin x) (x2/2) dx

Example 2: Find ∫ log x dx

Solution: Do you know any function whose derivative is log x? Guessing it is difficult. Hence, let’s take the first function f(x) = log x. So, the second function g(x) = 1. And, we know that ∫ 1 dx = x. Therefore, ∫ g(x) dx = x. Therefore, we have,

Example: 

Solution: