Expression Substitution

Expression Substitution is useful when we have to convert a very complicated function into a smple form. We can use Expression Substitution in inverse trigonometric function, differentiation of function, and sometimes even in the integration of function. If we find any of the expressions involved in function, we first need to make the right substitution and then proceed. If no such expression is found, we can also do the mathematical operation without substitution.

Expression Substitution:

The following are some of the most useful substitutes for simplifying mathematical expressions and making the differentiation and integration process easier.

x = aSinθ or x = aCosθ is used for the  function f(a2−x2)

x = a Secθ or x = aCosecθ is used for the function f(x2−a2)

x = aTanθ, or x = aCotθ is used for the function f(x2+a2). f(a2+x2) 

x = a Cos2θ is used for the functions f( a-xa+x ), f( a+xa-x )

x= atan θ  is used for the functions  f( a-xa+x ), f( a+xa-x )

x= atan θ  is used for the functions  f( 2×1+x2 ), f2x1-x2 

a = rcosα, b = rsinα is used for the functions  f(x) = acosx + bcosx

x = a sin2 θ + b cos2 θ  is used for the functions  f(x) = x -α or ꞵ -x

x = a(1 – cos θ)is used for the function f(2ax−x2)

Integration by Substitution:

Integration by substitution is utilised when the integration of a given function cannot be accomplished directly because the algebraic function is not in the standard form. Furthermore, the provided function can be reduced to its standard form by substituting appropriately. 

Consider the indefinite integral ∫f(x).dx, of a function f(x), for evaluation. By replacing x with g(t) and substituting, this integral can be changed into another form, as follows:

x=g(t).

I = ∫f(x).dx

x = g(t) where dx/dt = g'(t)

dx = g'(t).dt

I =∫f(x).dx=∫f(g(t)).g′(t).dt

Performing Substitutional Indefinite Integration:

The instructions below will help you in completing this approach of integration by substitution.

Step 1: For the given function to be reduced, choose a new variable, t.

Step 2: Where f(x) is integrated with respect to x, find the value of dx for the given integral.

Step 3: Make the necessary changes to the function f(x), as well as the new value dx.

Step 4: Integrate the function that you got from the substitution.

Step 5: To get the final solution, replace the original variable x in the final answer.

Here is how it is done,

To find I=∫f(x)dx

Take x=g(t), after differentiating dxdt=g'(t)dx=g'(t)dt

Substituting values I=∫f(x)dx=∫f(g(t))g'(t)dt

Let us help you understand this in another way,

The chain rule for derivatives corresponds to substitution for integrals.

Take F(u)as antiderivative of f(u):

∫f(u)du=F(u)+c

Take u=u(x)is a differentiable function, and the chain rule is applied

ddx∫F(u(x))=F'(u(x))u'(x)=f(u(x))u'(x)

Integrating both sides

∫f(u(x))u'(x)dx=F(u(x))+c

∫f(u(x))u'(x)dx=∫f(u)du, where u=u(x)

Numericals related to Expression Substitution:

1.  ∫etan-1×1+x2dx

Solution: Expression is: ∫etan-1×1+x2dx

Take tan-1x=t and differentiate it: ddxtan-1x=dtdx

11+x2=dtdx

dx=(1+x2)dt

Substitute the value:

∫etan-1×1+x2dx=∫et(1+x2)1+x2dt=∫etdt=et+c=etan-1x+c

Let’s find the integral of 2xsec2(x2+1)

Expression is: ∫2xsec2(x2+1)dx

Take x2+1=t and differentiate it: ddx(x2+1)=dtdx

2x=dtdxdx=dt2x

Substitute the value:

∫2xsec2(x2+1)dx=∫2xsec2(x2+1)dt2x=∫sec2tdt=tant+c=tan(x2+1)+c

Conclusion :

Substitution is a technique for simplifying a system of equations by expressing one variable in terms of another and thereby eliminating one variable from the equation. Then, calculate this equation and replace it till you find the solution. Expression Substitution aims to simplify equations by rewriting them in terms of a single variable. The crucial point to remember here is that you are constantly substituting equal values.