   Chapter 4.3, Problem 56E

Chapter
Section
Textbook Problem

# Find the derivative of the function. g ( x ) = ∫ tan x x 2 1 2 + t 4   d t [ 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 the function gx=tanxx212+t4 dt

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=tanxx212+t4 dt

4) Calculation:

Rewrite the given integral as (by using hint given)

tanxx212+t4 dt=tanx012+t4 dt+0x212+t4 dt

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

=-0tanx12+t4 dt+0x212+t4 dt

Differentiating with respect to x

g'x=ddx-0tanx12+t4 dt+ddx0x212+t4 dt

By using Fundamental theorem of Calculus, Part 1,

g'x=-12+t4ddtt- 12+t4ddt0+12+t4

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