MATHEMATICS  THEORY



In
indefinite integral section,
various formulas have been introduced to calculate the integral of a
function. However, it is not possible
to have a formula for every possible situation. Thus, basic formulas like
need to be generalized so they can be used in a variety of cases. This
rule is called the substitution rule. 





The Substitution Rule for Indefinite Integrals



The substitution rule states:
If u = g(x) is a differentiable function in an interval
R and f is continuous on R, then




Original Function f(x)
Substitution Function for f(x)


This rule can be better understand by calculating
.
Let u = 5x^{2} + 7. The derivative of u with respect to x is
du/dx = 10x
Rearrange the above equation gives
du = 10xdx
Thus






In order to double check the answer, the derivative of
function can
be taken.
The result confirms the correction of . 





The Substitution Rule for Definite Integrals



Substitution rule can also be used for definite integrals.
The substitution rule for definite integrals states:
If g'(x) is continuous on interval R and f(x) is continuous on the range
of g(x), then






This rule can be better understand by calculating
.
Let u = 5x^{2} + 7. The derivative of u with respect to x is
du/dx = 10x
Rearrange the above equation gives
du = 10xdx
The new limits of integration can be calculated.
When x = 0, u = 7 and when x = 1, u = 12. Therefore,
It is obviously that the substitution rule provides an easy way to calculate
the integral of a function. 


