   Chapter 4.3, Problem 54E

Chapter
Section
Textbook Problem

# Find the derivative of the function. g ( x ) = ∫ 1 − 2 x 1 + 2 x t sin t   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=1-2x1+2x tsint dt

Explanation

1) Concept:

i) Property 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=1-2x1+2x tsint dt

4) Calculation:

Rewritethe given integral as (by using the given hint)

1-2x1+2x tsint dt=1-2x0tsint dt+01+2xtsint dt

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

1-2x1+2x tsint dt=-01-2xtsint dt+01+2xtsint dt

Differentiating with respect to x

g'x=ddx1-2x1+2x tsint dt=ddx-01-2x tsint dt+ddx01+2x tsint dt

By usingFundamental theorem of Calculus, Part

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