554
Chapter 5 | Integration
Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of g ( r ) = ∫ 0 r
5.16
x 2 +4 dx .
Example 5.18 Using the Fundamental Theorem and the Chain Rule to Calculate Derivatives
x sin tdt . Find F ′( x ).
Let F ( x ) = ∫ 1
Solution Letting u ( x ) = x , wehave F ( x ) = ∫ 1 u ( x )
sin tdt . Thus, by the Fundamental Theorem of Calculus and the chain
rule,
⎝ u ( x ) ⎞ ⎠ du dx = sin( u ( x )) · ⎛ ⎝ 1 2
F ′( x ) = sin ⎛
x −1/2 ⎞ ⎠
= sin x
2 x .
x 3
5.17
Let F ( x ) = ∫ 1
cos tdt . Find F ′( x ).
Example 5.19 Using the Fundamental Theorem of Calculus with Two Variable Limits of Integration
2 x
Let F ( x ) = ∫ x
t 3 dt . Find F ′( x ).
Solution We have F ( x ) = ∫ x 2 x
t 3 dt . Both limits of integration are variable, so we need to split this into two integrals. We
get
2 x
F ( x ) = ∫
t 3 dt
x
0
2 x
= ∫
t 3 dt + ∫
t 3 dt
x
0
2 x
x
=− ∫
t 3 dt + ∫
t 3 dt .
0
0
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