The determinant is a scalar value that can be obtained from the elements of a square matrix.
Integration is a method of addition of slices to calculate the whole. It is the reverse process of finding the derivative of a function. When we differentiate a function, it shows the rate of change of a function at a given point. For real-valued functions, it is the slope of the tangent line at a point on a graph.
Integration is an important topic for the JEE exam. Students can expect 2-3 questions from this topic. Students can easily score marks from this topic if they are thorough with the concept.
In this article, we will have a look at the integration of determinants.
It is an arrangement of numbers in the form.
The concept of determinants originated when mathematicians were trying to solve a system of simultaneous linear equations.
The horizontal lines are the rows and vertical lines are the columns.
Here D = ad – cb.
For a 3×3 matrix, the determinant is given by
= a1(b2c3-b3c2) – b1(a2c3-a3c2) + c1(a2b3-a3b2).
The shape of every determinant will be a square. If a determinant contains n rows and n columns, then it is of order n.
Integration Of Determinants
Let f(x), g(x) and h(x) are functions of x. Let a, b, c. α, β and γ are constants such that
The integral of ∆(x) is given by
We integrate the functions f(x), g(x) and h(x). Then we find the determinant.
If the elements of more than one row or column are functions of x, then we do integration after evaluating the expansion of the determinant.
Properties of Determinants
- The value of the determinant remains unchanged if the rows and columns are interchanged.
- The sign of the determinant changes If any two rows or columns of a determinant are interchanged.
- When any two rows or columns of a determinant are the same, then the determinant is 0.
- When any row or column of the determinant is multiplied by a variable k, then its value is multiplied by k.
- When some or all elements of a row or column are expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
Differentiating a function means to find the rate of change. It is an important topic for the JEE exam. The derivative of a function y = f(x) of a variable x is the rate of change of y with respect to the rate of change of x. It is written as dy/dx. In mathematical terms, it is written as
As far as the JEE exam is concerned, derivatives, integration, and determinants have great importance. Both integration and differentiation are inverse processes of each other. The derivative of any function will be unique but, the integral of every function is not unique. Two integrals of the same function will differ by a constant.
Some standard derivatives are given below.
- (d/dx)xn = nxn-1
- (d/dx) c = 0, c is a constant
- (d/dx) sin x = cos x
- (d/dx) cos x = -sin x
- (d/dx) tan x = – sec2 x
- (d/dx) log x = 1/x