   Chapter 5.3, Problem 61E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the derivative of the function. F ( x ) = ∫ x x 2 e t 2   d t

To determine

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

Explanation

Given information:

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.

F(x)=xx2et2dt (1)

Here, F(x) is function of x and t is a derivative function.

Consider t1=x (2)

Differentiate both sides of the Equation (2).

dt1dx=ddx(x)=1

Consider v=x2 (3)

Differentiate both sides of the Equation (3).

dt2dx=ddx(x2)=2x

Modify the Equation (1) as shown in below.

F(x)=x0et2dt+0x2et2dt (4)

Reverse the integral using equation (4).

F(x)=0xet2dt+0x2et2dt (5)

Substitute t1 for x and t2 for x2 in Equation (3)

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