Definite integral of these questions
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Definite integral of these questions

[From: ] [author: ] [Date: 11-04-27] [Hit: ]
u = 0 and when x = e,let 1/x = u, when x = 1, u = 1 and when x = 2,= 1.-e^(1/2)+e = e-sqrt(e)-plug,......
dx/x(1-ln^2 x))^1/2 from 1 to e
and
(e^1/x)dx/x^2 from 1 to 2

-
∫dx /x√1 - (ln x)^2] from 1 to e

let ln x = u, when x = 1, u = 0 and when x = e, u = 1
dx/x = du

∫ du /√(1 - u^2) from 0 to 1

= sin^-1(u) from 0 to 1

= Π/2 - 0

= Π/2

2)

∫e^(1/x)dx /x^2 from 1 to 2

let 1/x = u, when x = 1, u = 1 and when x = 2, u = 1/2
-dx/x^2 = du
dx/x^2 = -du

- ∫e^(u) du from 1 to 1/2

= -e^u from 1 to 1/2

= - [e^(1/2) - e^1 ]

= e - e^(1/2)

= 1.0696

-
1) INTEGRAL[dx/[x(1-ln^2(x))^1/2]: 1 to e]
u=ln(x) then du=dx/x
INTEGRAL[1/sqrt(1-u^2) du]
sin-1(u)+C
sin-1(ln(x)):1 to e
=sin-1(1)=pi/2
#2) INTEGRAL[e^(1/x)/x^2 dx:1 to 2]
u=1/x and du=-1/x^2 then -du=1/x^2
INTEGRAL[-e^u du]
-e^u+C
-e^(1/x):1 to 2
-e^(1/2)+e = e-sqrt(e)

-
plug, use fnInt in your calculator
1
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