Integration by Substitution

Let’s learn what is Integration before understanding the concept of Integration by Substitution. The integration of a function f(x) is given by F(x) and it is represented by:

∫f(x)dx = F(x) + C

Here R.H.S. of the equation means integral of f(x) with respect to x.

• F(x) is called anti-derivative or primitive.
• f(x) is called the integrand.
• dx is called the integrating agent.
• C is called constant of integration or arbitrary constant.
• x is the variable of integration.

The anti-derivatives of basic functions are known to us. The integrals of these functions can be obtained readily. But this integration technique is limited to basic functions and in order to determine the integrals of various functions, different methods of integration are used. Among these methods of integration let us discuss integration by substitution.

Integration by Substitution Method

In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others.

Take for example an equation having an independent variable in x, i.e. ∫sin (x³).3x².dx———————–(i),

In the equation given above the independent variable can be transformed into another variable say t.

Substituting x³ = t ———————-(ii)

Differentiation of above equation will give-

3x².dx = dt ———————-(iii)

Substituting the value of (ii) and (iii) in (i), we have

∫sin (x³).3x².dx = ∫sin t . dt

Thus the integration of the above equation will give

∫sin t . dt= -cos t + c

Again putting back the value of t from equation (ii), we get

∫sin (x³).3x².dx = -cos x³ + c

The General Form of integration by substitution is:

∫ f(g(x)).g'(x).dx = f(t).dt, where t = g(x)

Usually the method of integration by substitution is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. Doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function.

When to Use Integration by Substitution Method?

In calculus, the integration by substitution method is also known as the “Reverse Chain Rule” or “U-Substitution Method”. We can use this method to find an integral value when it is set up in the special form. It means that the given integral is of the form:

∫ f(g(x)).g'(x).dx = f(u).du

Here, first, integrate the function with respect to the substituted value (f(u)), and finish the process by substituting the original function g(x).

Integration by Substitution Example

To understand this concept better, let us look into the examples.

Example 1:

Find the integration of

Solution:

Given :

Let t = tan^-1x …… (1)

dt = (1/ 1+x² ) . dx

I = ∫ e^t . dt

= e^t + C …….(2)

Substituting the value of (1) in (2), we have I = e^tan-1x + C. This is the required integration for the given function.

Example 2:

Integrate 2x cos (x² – 5) with respect to x .

Solution:

I = ∫2xcos(x² – 5).dx

Let x² – 5 = t …..(1)

2x.dx = dt

Substituting these values, we have

I = ∫cos(t).dt

= sin t + c …..(2)

Substituting the value of 1 in 2, we have

= sin (x² – 5) + C

This is the required integration for the given function.
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