Question

Calcule a integral indefinida ∫cosx /(sen^2 x) dx.

Qual a resposta??
a)ln|sen x|+ c
b)ln|cos⁡x |+ c
c)sec x + c
d)- cossec x + c
e)- cotg x + c

Answer 1

Calcular a integral indefinida

     $\displaystyle\int\frac{\cos x}{\mathrm{sen^2\,}x}\,dx\\\\\\ =\int\frac{1}{(\mathrm{sen\,}x)^2}\cdot \cos x\,dx$


Faça uma substituição:

     $\mathrm{sen\,}x=u\quad\Rightarrow\quad \cos x\,dx=du$


e a integral fica

     $\displaystyle=\int\frac{1}{u^2}\,du\\\\\\ =\int u^{-2}\,du\\\\\\ =\frac{u^{-2+1}}{-2+1}+C\\\\\\ =\frac{u^{-1}}{-1}+C\\\\\\ =-\,\frac{1}{u}+C\\\\\\ =-\,\frac{1}{\mathrm{sen\,}x}+C$

     $=-\,\mathrm{cossec\,}x+C$     <————    esta é a resposta.


     Resposta:  alternativa  d)  − cossec x + C.


Bons estudos! :-)