Скачать или смотреть D.I. Method |Integration By parts|IIT Jee maths|CBSE BOARD|ICSE BOARD|UP BOARD

D.I. Method |Integration By parts|IIT Jee maths|CBSE BOARD|ICSE BOARD|UP BOARD
Integration by parts is a calculus technique to find the integral of a product of two functions, using the formula ∫uv dx = u∫v dx – ∫(u'∫v dx) dx. It's derived from the product rule for differentiation and allows you to transform a difficult integral into potentially simpler ones. The key is choosing the correct "u" and "dv" (where u' is the derivative of u, and v is the integral of dv) using the ILATE (or LIATE) rule to simplify the integral term.
This video explains the concept of integration by parts with examples:
The Formula and Its Components
The core of integration by parts is this formula: ∫uv dx = u∫v dx – ∫(u'∫v dx) dx
u: The first function.
v: The second function.
∫v dx: The integral of the second function.
u': The derivative of the first function (du/dx).
This video shows how to apply the integration by parts formula:
The ILATE Rule for Choosing 'u' and 'v'
Choosing the correct "u" and "v" is crucial for the method to work effectively. The ILATE (or LIATE) rule provides a helpful guideline:
I: nverse trigonometric functions (e.g., arcsin(x), arctan(x))
L: ogarithmic functions (e.g., ln(x))
A: lgebraic functions (e.g., x, x², 5)
T: rigonometric functions (e.g., sin(x), cos(x))
E: xponential functions (e.g., eˣ)
You should choose 'u' to be the function that comes first in this order. The remaining part of the integrand becomes 'dv'.
How it Works
Identify u and dv: Choose the functions from your integrand for 'u' and 'dv' using the ILATE rule.
Differentiate u: Find the derivative of 'u' to get u'.
Integrate dv: Find the integral of 'dv' to get v.
Apply the formula: Substitute u, v, u', and ∫v dx into the integration by parts formula.
Simplify: The goal is that the new integral (∫u'∫v dx dx) is simpler to solve than the original integral.
Example
To integrate x sin(x):
Choose u and dv: According to ILATE, you'd choose u = x (Algebraic) and dv = sin(x) dx (Trigonometric).
Find u' and v: u' = 1, and v = ∫sin(x) dx = -cos(x).
Apply the formula:
∫x sin(x) dx = x(-cos(x)) – ∫(-cos(x))(1) dx
Simplify:
-x cos(x) + ∫cos(x) dx