Indirect Substitution In Integral

Introduction

An infinite integral is a function that takes the antiderivative of another function. It’s represented by an integration sign, a function, and dx at the end. The antiderivative can be represented more simply using the indefinite integral. The following are the three ways for determining the value of an indefinite integral:

• Substitutional integration [direct and indirect]
• Partial fractions integration
• Parts-based integration

Indirect Substitutions

• If the integrand of f (x) is the product of the functions f (x) and h (x), with h (x) being the function of the integral f (x), the integral f (x) may be represented as t.
• If any function ∫ f (x) dx = g (x), then ∫ f (ax + b) = [1 / a] g (ax + b)
• The integral of the form ∫ [f’ (x) / f (x)] dx = log (f (x)) + c
• The integral ∫ [f’ (x) / √ f (x)] dx = 2 √[f (x)] + c
• ∫ [[f (x)]ⁿ f’ (x)] dx = {[f (x)]ⁿ⁺¹ / (n+1) + C}, where n ≠ 1

The identities of trigonometry are used to construct integrals of the type ∫sin^px dx, ∫cos^qx dx [p less than or equal to 4]. sin^px and cos^qx are stated in terms of sines and cosines of multiples of x.

The steps for evaluating integrals of the type sin^px and cos^qxdx, are as follows:

1] Verify that the sinx and cosx exponents belong to the set of natural numbers.

2] The following are the various exponent cases:

a] Assume sinx exponent is odd, then cosx = t.

b] If cosx exponent is odd, then sinx = t is used.

c] If both exponents are odd, the variable “t” is considered to be sinx or cosx.

d] When both exponents are positive, the trigonometric identities are used to represent them in terms of sines and cosines of x.

The Substitution Method

A given integral f(x) dx can be changed into a different form using the substitution method by replacing the independent variable x with t. This is done by inserting x = g(t) into the equation.

Consider, I = ∫ f(x) dx
Substitute x = g(t) for dx/dt = g'(t) or dx = g'(t)dt to get dx = g'(t)dt.
Therefore, I = ∫ f(x) dx = ∫ f[g(t)] g’(t)dt

FAQ’S

What is the mechanism of substitution?

It's a technique for simplifying a system of equations by expressing one variable in terms of another and therefore eliminating one variable from the equation. Then, calculate this equation and back replace it till you get the solution.

Ques 2: How do you go about using the substitution method?

Ans 2. Follow the instructions below to solve a system of equations:

Begin by solving one equation for one variable (in y = or x = form, for example).
Solve the other problem by substituting this expression for the missing variable.
Then, in order to obtain an answer, you must substitute your solution into the first equation and solve it.

Ques 3: What is the best way to apply the substitution approach when there are two variables?

Ans 3: To solve for two variables, use the following steps:

To begin, pick a single equation and solve for one of its variables.

Then, in the other equation, substitute the variable you just solved.

Solve the new equation now.

Finally, solve for the other variable by substituting the value discovered into any equation.