FIRST FUNDAMENTAL THEOREM OF CALCULUS - PART 1

The Fundamental Theorem of Calculus (Part-1) is an extremely powerful theorem that establishes the relationship between differentiation and integration.

First Fundamental Theorem of Calculus (Part - 1)

If f(x) is continuous over an interval [a, b], and the function F(x) is defined by

Then the derivative of F(x) is

F'(x) = (x)'f(x) - (a)'f(a)

Using the First Fundamental Theorem of Integral Calculus, find the derivative of g(x) in each case.

Example 1 :

Solution :

Example 2 :

Solution :

Example 3 :

Solution :

Example 4 :

Solution :

g'(x) = (x3)'(cos x3) - (1)'(cos 1)

g'(x) = (3x2)(cos x3) - (0)(cos 1)

g'(x) = 3x2cos x3

Example 5 :

Solution :

g'(x) = (2x)'[(2x)3] - (x)'(x3)

g'(x) = (2)(8x3) - (1)(x3)

g'(x) = 16x- x3

g'(x) = 15x3

Example 6 :

Solution :

g'(x) = (x3)'(cos x3) - (x)'(cos x)

g'(x) = (3x2)(cos x3) - (1)(cos x)

g'(x) = 3x2cos x3 - cos x

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