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).