Integral Definida

2 participantes

PiR2 :: Matemática :: Iniciação ao Cálculo

Página 1 de 1

Integral Definida

por Giiovanna Ter 03 Set 2013, 11:59

Calcule

R: pi/2

Minha integral indefinida deu 1/2 arctan( 2 tan(x/2) ), mas não estou conseguindo chegar no gabaritp.

Giiovanna: Grupo
Velhos amigos do Fórum; Mensagens : 2128
Data de inscrição : 31/08/2012
Idade : 30
Localização : São Paulo, SP

$Integral Definida Gif.latex?%5Chspace%7B-16%7D%24Using%20the%20formula%20%24%5Cbf%7B%5Cint_%7B0%7D%5E%7B2a%7Df%28x%29dx%20%3D%20%5Cint_%7B0%7D%5E%7Ba%7Df%28x%29dx%5C%3B%20%2C%5C%3B%7D%24%20If%20%24%5Cbf%7Bf%282a-x%29%3Df%28x%29%7D%24%5C%5C%5C%5C%5C%5C%20Let%20%24%5Cbf%7BI%20%3D%5Cint_%7B0%7D%5E%7B2%5Cpi%7D%5Cfrac%7B1%7D%7B5-3%5Ccos%20x%7Ddx%3D2%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7B1%7D%7B5-3%5Ccos%20x%7Ddx%7D%24%5C%5C%5C%5C%5C%5C%20bcz%20%24%5Cbf%7Bf%282%5Cpi-x%29%3D%5Cfrac%7B1%7D%7B5-3%5Ccos%20%282%5Cpi-x%29%7D%3D%5Cfrac%7B1%7D%7B5-3%5Ccos%20x%7D%3Df%28x%29%7D%24%5C%5C%5C%5C%5C%5CLet%20%24%5Cbf%7Bco%20x%3D%20%5Cfrac%7B1-%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B1+%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7D%24%5C%5C%5C%5C%5C%5C%20%24%5Cbf%7B%3D2%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7B1+%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B5%5Cleft%281+%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%5Cright%29-3%5Cleft%281-%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%5Cright%29%7Ddx%7D%24%5C%5C%5C%5C%5C%5C%20%24%5Cbf%7B%3D2%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7B%5Csec%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7D%7B2+8%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D%7Ddx%7D%24%5C%5C%5C%5C%5C%5C%20%24%5Cbf%7B%3D%5Cfrac%7B2%7D%7B8%7D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7B1%7D%7B%5Ctan%5E2%20%5Cfrac%7Bx%7D%7B2%7D+%5Cfrac%7B1%7D%7B2%5E2%7D%7Ddx%7D%24%5C%5C%5C%5C%5C%5C%20Now%20Let%20%24%5Cbf%7B%5Ctan%20%5Cfrac%7Bx%7D%7B2%7D%20%3D%20t%5CLeftrightarrow%20%5Csec%5E2%20%5Cfrac%7Bx%7D%7B2%7Ddx%20%3D%202dt%7D%24%5C%5C%5C%5C%5C%5C%20%24%5Cbf%7B%3D%5Cfrac%7B4%7D%7B8%7D.%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7Bt%5E2+%5Cfrac%7B1%7D%7B2%5E2%7D%7Ddt%7D%24%5C%5C%5C%5C%5C%5C%20%24%5Cbf%7B%3D%5Cfrac%7B1%7D%7B2%7D.%5Cfrac%7B2%7D%7B1%7D.%5Cleft%5B%5Ctan%5E%7B-1%7D%28%5Cinfty%29-%5Ctan%5E%7B-1%7D%280%29%5Cright%5D%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Capprox%201$