Applied Calculus: UNIT III: Integral Calculus
Net Change Theorem: Integration by parts
Method of integration
Explanation, Formula, Equation, Example and Solved Problems - Integral Calculus: 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)
(b) INTEGRATION BY PARTS
Every differentiation
rule has a corresponding integration rule. For instance, the substitution rule
for integration corresponds to the chain rule for differentiation. The rule
that corresponds to the product rule for differentiation is called the rule for
integration by parts.
The product rule states
that if f and g are differentiable
functions, then
d/dx [f(x)g(x)] = f(x)g'(x) + g(x)f '(x)
In the notation for
indefinite integrals this equation becomes,
∫[f(x)g'(x) + g(x)f'(x)] dx
= f(x)g(x)
∫ f(x)g'(x)dx + ∫g(x)f'(x)dx
= f(x)g(x)
∫f(x)g'(x)dx = f(x)g(x) ‒ ∫g(x)f'(x)dx
The Formula in Equation
(*) where is the equation is called the formula for integration by parts. The
above formula can be stated by making use of the substitution u = f(x), v = g(x), du = f'(x) dx and dv = g'(x)dx as given
below:
∫udv = uv – ∫vdu
Example
46. Find ∫x cos5x dx
Solution:
Applied Calculus: UNIT III: Integral Calculus : Tag: Applied Calculus : Method of integration - Net Change Theorem: Integration by parts
Applied Calculus: UNIT III: Integral Calculus
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Applied Calculus
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