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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the derivative of the function.

g ( x ) = 1 2 x 1 + 2 x t sin t d t

To determine

the derivative of the function g(x) by using the theorem of fundamental calculus.

Explanation

Given information:

Show the theorem of fundamental calculus (part 1) as shown in below.

F(x)=axf(t)dtF(x)=f(x)

The theorem of fundamental calculus (part 1) is valid, if f is continuous on [a,b].

Calculation:

Show the integral function as in below.

g(x)=12x1+2xtsintdt (1)

Here, g(x) is function of x and u is a derivative function.

Consider u=12x (2)

Differentiate both sides of the Equation (2).

u=12xdudx=2

Consider v=1+2x (3)

Differentiate both sides of the Equation (3).

v=1+2xdvdx=2

Modify Equation (1) by applying chain rule as shown in below.

g(x)=12x0tsintdt+01+2xtsintdt (4)

Reverse the limits of integral using equation (2).

g(x)=012xtsintdt+01+2xtsintdt (5)

Applying fundamental calculus part 1 as shown in below

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