Dr. Brajesh Kumar Jha

📥 Download Partial Fractions PDF Integration by Partial Fractions — Examples & Solutions Integration by Partial Fractions Types and worked examples (converted from the uploaded notes). Type 1: Distinct linear factors When the denominator is expressible as a product of distinct linear factors. Example 1 Evaluate \( \displaystyle \int \frac{x-1}{(x+1)(x-2)} \, dx \). Write \( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{A}{x+1} + \dfrac{B}{x-2} \) Then \( x-1 = A(x-2) + B(x+1) \) Putting \(x=2\): \(1 = 3B \Rightarrow B=\tfrac{1}{3}\). Putting \(x=-1\): \(-2 = -3A \Rightarrow A=\tfrac{2}{3}\). Thus \( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{2/3}{x+1} + \dfrac{1/3}{x-2} \) Integrate: \( \int \frac{x-1}{(x+1)(x-2)} dx = \frac{2}{3}\ln|x+1| + \frac{1}{3}\ln|x-2| + C \) Example 2 Evaluate \( ...

Types of Matrices — Definitions & Examples Types of Matrices — Definitions & Examples A concise reference listing common matrix types with a short definition and two examples for each. Perfect for students, teachers, and blog readers. Table of contents Row Matrix Column Matrix Square Matrix Rectangular Matrix Zero (Null) Matrix Identity Matrix Diagonal Matrix Scalar Matrix Triangular Matrices Symmetric Matrix Skew-Symmetric Matrix Singular Matrix Non-Singular Matrix Orthogonal Matrix Sparse Matrix Row Matrix Definition: A matrix with only one row. Examples [ 3 5 7 ] [ -2 4 9 1 ] Column Matrix Definition: A matrix with only one column. Examples [ 4 -1 2 ]...

  Integration by Part If u and v are two functions of x, then Remark: The proper choice of u and v is very important to apply integration by part. We can choose the first function (u) as the function which comes first in the word ILATE. Here I stands for the inverse trigonometric function (Ex. ) L stands for logarithmic function (Ex. ) A stands for the algebraic functions (Ex. ,   etc.) T stands for the trigonometric functions (Ex. ) E stands for the exponential functions. (Ex. ) Example 1: Evaluate Solution: Here as per the above remark ILATE   (Algebraic function) comes first and   (trigonometric function). Thus let   and . Now by formula Example 2: Evaluate   Solution: Here as per the above remark ILATE   (Algebraic function) comes first and   (trigonometric function). Thus let   and . Now by formula Where   and Here in second part of integration ( ) we will apply in...