   Chapter 4.3, Problem 53E

Chapter
Section
Textbook Problem

# Find the derivative of the function. g ( x ) = ∫ 2 x 3 x u 2 − 1 u 2 + 1   d u [ H i n t : ∫ 2 x 3 x f ( u )   d u = ∫ 2 x 0 f ( u ) d u + ∫ 0 3 x f ( u )   d u ]

To determine

To find:

The derivative of function gx=2x3xu2-1u2+1 du

Explanation

1) Concept:

i) Properties of definite integral:

ba fxdx=-ab fxdx

2) Fundamental theorem of Calculus, Part 1:

If f is continuous on a, b, then the function g defined by

gx=0x ft dt   axb

gx is continuous on a,b, and g'x=f(x)

3) Given:

gx=2x3xu2-1u2+1 du

4) Calculation:

Rewrite given integral as (by using hint given),

2x3xu2-1u2+1 du=2x0u2-1u2+1 du+03xu2-1u2+1 du (1)

Now by using concept i) (Property of definite integral),

=-02xu2-1u2+1 du+03xu2-1u2+1 du

gx=-02xu2-1u2+1 du+03xu2-1u2+1 du

Differentiating with respect to x,

g'x=ddx-02xu2-1u2+1 du+ddx03xu2-1u2+1 du

By using Fundamental theorem of Calculus, Part 1,

g'x=-u2-1u2+

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