Question

O valor da integral definida de 0 a x (cos x/3 + sen x/2)dx é?

Answer 1

$\int\limits^x_0 ({ \frac{cosx}{3}+ \frac{sen x}{2} }) \, dx$
Separando as integrais da soma é a soma das integrais

$\int\limits^x_0 { \frac{cos x}{3} } \, dx + \int\limits^x_0 { \frac{senx}{2} } \, dx$

Retirando as constantes para fora da integral

$\frac{1}{3} \int\limits^x_0 { cosx } \, dx + \frac{1}{2} \int\limits^x_0 { sen x } \, dx$

$\frac{1}{3}[senx]^x_{0} + \frac{1}{2}[-cosx]^x_{0}$

$= \frac{1}{3}[senx - sen0] - \frac{1}{2}[cosx - cos0]$

$= \frac{1}{3}[senx - 0] - \frac{1}{2}[cosx - 1]$

$= \frac{senx}{3} - \frac{cosx}{2} + \frac{1}{2}$