Applied Calculus: UNIT III: Integral Calculus
The indefinite integrals
Explanation, Formula - The indefinite integrals
THE
INDEFINITE INTEGRALS
Both parts of the
fundamental theorem establish connection between anti derivatives and definite
integrals. Part 1 says that if is f is
continuous then a∫xf(t)dt
is an anti derivative of f part 2
says that a∫bf(t)dt
evaluating F(b) ‒ F(a), where F is anti derivative of f.
We need a convenient
notification for anti‒derivatives that makes them easy to work with. Because of
the relation given by the fundamental theorem between anti derivatives and
integrals. the notation ∫ f(x) dx is
traditionally used for an anti derivative of f and is called an infinite
integrals.
Thus ∫ f(x) dx = F(x) = f(x) we should
distinguish carefully between definite and indefinite integrals. A definite
integrals a∫b f(t)dt
is a number, where as an indefinite integrals ∫ f(x) dx = F(x) is a
function (or family of functions).
The connection between
them is given by part 2 of the fundamental theorem
If f is continuous on [a, b], then
The effectiveness of
the fundamental theorem depends on the having a supply of anti‒derivatives of
functions.
For instances any
formula can be verified by differentiating the function on the right side and
obtaining the integrand
∫ sec2x dx =
tan x + c because
d/dx (tan x + c) = sec2x
Part 2 of the
fundamental theorem says that if f is
continuous on [a, b], then So a∫b f (t)dt = F(b) − f(a) where F is a anti derivatives of f. This means that F ' = f so the equation can be rewritten as a∫b
f(t)dt = F(b) ‒ f(a). we know that
F' represents the rate of change of y = F(x)
with repeat to x and F(b) ‒ F(a) is the changes in y when x changes
from a to b This concept is started as a theorem called the net charges theorem
Applied Calculus: UNIT III: Integral Calculus : Tag: Applied Calculus : - The indefinite integrals
Applied Calculus: UNIT III: Integral Calculus
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