Como calcular essa integral?

por GercinoNogueira Sex 26 Set 2014, 13:43

Resposta:
$Como calcular essa integral? Gif$

por GercinoNogueira Seg 29 Set 2014, 14:08

Como a resolução é gigante, vou pôr passo-a-passo aqui. Primeiro, lembre-se que tg²x + 1 = sec²x:
$Como calcular essa integral? Gif$

por GercinoNogueira Seg 29 Set 2014, 14:13

Chamaremos u = tgx, a gente faz du = sec²dx, logo dx = du/sec²x:
$Como calcular essa integral? Gif$

por GercinoNogueira Seg 29 Set 2014, 14:34

Lembrando que sec²x=1/cos²x e que cos²x = 1/2 + cos2x/2, continuaremos os cálculos: $Como calcular essa integral? Gif.latex?%3D%5Cfrac%7Bu%5E%7B9%7D%7D%7B9%7D+%5Cfrac%7Bu%5E%7B7%7D%7D%7B7%7D%3D%5Cfrac%7Btg%5E%7B9%7Dx%7D%7B9%7D+%5Cfrac%7Btg%5E%7B7%7Dx%7D%7B7%7D%3D%5Cfrac%7B7tg%5E%7B9%7Dx+9tg%5E%7B7%7Dx%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%287tg%5E%7B2%7Dx+9%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%5B7%28sec%5E%7B2%7Dx-1%29+9%5D%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%287sec%5E%7B2%7Dx-7+9%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%287sec%5E%7B2%7Dx+2%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%287%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx%7D+2%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%287%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx%7D+2%5Cfrac%7Bcos%5E%7B2%7Dx%7D%7Bcos%5E%7B2%7Dx%7D%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%5Ccdot%20%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx%7D%287+2%5Ccdot%20cos%5E%7B2%7Dx%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%5Ccdot%20sec%5E%7B2%7Dx%5B7+2%28%5Cfrac%7B1%7D%7B2%7D+%20%5Cfrac%7Bcos2x%7D%7B2%7D%29%5D%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%5Ccdot%20sec%5E%7B2%7Dx%287+1+cos2x%29%7D%7B63%7D%3D%5Cfrac%7Btg%5E%7B7%7Dx%5Ccdot%20sec%5E%7B2%7Dx%288+cos2x%29%7D%7B63%7D$