Applied Calculus: UNIT III: Integral Calculus
Net Change Theorem: Substitution rule
Method of integration
Explanation, Formula, Equation, Example and Solved Problems - Net Change Theorem : Method of integration : Substitution rule
METHOD
OF INTEGRATION
Corresponding to the
various rules in the different calculus for differentiating sum, product and
functions of functions, more or less there are similar rules available in the
integral calculus also.
(a) Substitution rule
(b) Integration by
parts
(c) Successive
reduction method
(d) Decomposition into
a sum (Method of partial fractions)
(a) SUBSTITUTION RULE
Because of the
fundamental theorem, its important to be able to find anti‒ derivatives. but
anti differentiation formulas don't tell us how to evaluate integrals of
certain forms therefore we have to identify
a suitable method to find the anti‒derivative of the integrand one of such
method is the Substitution method and described as given below.
INTEGRALS OF FUNCTIONS CONTAINING LINEAR FUNCTIONS OF x
Let ax + b = u. Then a
dx = du.
∫ f(ax
+ b) dx = 1/a f(u) du
Integrals
of functions of the form ∫ f(xn)
xn‒1dx
In this type put xn
= t then nxn‒1dx = dt
xn‒1dx = 1/n
dt.
The given integral
reduces to the form 1/n ∫ f(t)dt
which can be evaluated easily.
Integrals
of functions of the form ∫ [f(x)]n
f '(x) dx
It is given that ∫[f(x)]n f '(x) dx. Then put f(x) =
t, ⇒ f '(x)dx = dt.
The given integral is
evaluated depending on n.
= log(t) = log[f(x)] + C when n = ‒1.
Integrals
of functions of the form ∫ f[g(x)]g'(x)
dx
If u = g(x) is a
differentiable function whose range is an interval I and f is continuous on I, then
This rule is known as
the substitution rule.
Applied Calculus: UNIT III: Integral Calculus : Tag: Applied Calculus : Method of integration - Net Change Theorem: Substitution rule
Applied Calculus: UNIT III: Integral Calculus
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Applied Calculus
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