Integration by Substitution

In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable with others. Let’s consider an example for better understanding.

Example: Simplify ∫ 3x² sin (x³) dx.

Answer:

Let I = ∫ 3x² sin (x³) dx.

In order to evaluate the given integral lets substitute any variable by a new variable as:

Let x³ be t for the given integral.

Then, dt = 3x² dx

Therefore, 

I = ∫ 3x² sin (x³) dx = ∫ sin (x³) (3x² dx)

Now, substitute t for x³ and dt for 3x² dx in the above integral.

I = ∫ sin (t) (dt)

As ∫ sin x dx = -cos x + C, thus

I = -cos t + C

Again, substitute back x³ for t in the expression as:

I = ∫ 3x² sin (x³) dx = -cos x³ + C

Which is the required integral.

Hence, the General Form of integration by substitution is:

∫ f(g(x)).g'(x).dx = f(t).dx

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. By doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function.

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:

Read More,

• Calculus in Maths
• Integrals
• Integral Calculus
• Differentiation of Trig Functions
• Trigonometric Equations

Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.

Below is an example of the Integration of a given function.

e.g., Consider a function, f(y) = y². 

This function can be integrated as: 

∫y²dy = [Tex]\frac{y^{2+1}}{2+1}~+~C[/Tex]

However, an indefinite integral is a function that takes the anti-derivative of another function. It is represented as an integral symbol (∫), a function, and a derivative of the function at the end. The indefinite integral is an easier way to symbolize an anti-derivative.

Let’s learn what is integration mathematically, 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 and x is the variable of integration.