Question

A integral indefinida da função f(x)=sin(x)−tan(x) é dada por:

A)−cos(x)−ln∣cos(x)∣+C
B)sin(x)+ln∣tan(x)∣+C
C)−cos(x)−ln∣cos(x)4∣+C
D)−cos(x)+ln∣cos(2x)∣+C
E) −sin(x)+ln∣cos(x)∣+C

Answer 1

$\displaystyle \sf \int \left[sin(x)-tan(x)\right]dx \\\\\\ \int \left[sin(x)-\frac{sin(x)}{cos(x)}\right]dx \\\\\\ \int \left[\frac{sin(x)\cdot cos(x)-sin(x)}{cos(x)}\right]dx \\\\\\ \int\left[\frac{-sin(x)(1-cos(x))}{cos(x)}\right]dx \\\\\\ Fa{\c c}amos } : \\\\ cos(x) =u \to -sin(x)dx=du \\\\ Da{\'i}} : \\\\ \int \left[\frac{du(1-u)}{u }\right]dx \to \int \left(\frac{1}{u}-1\right)du \\\\\\ \int \frac{1}{u}du -\int du = \ln|u|-u+C \\\\\\ \huge\boxed{\sf -cos(x)+\ln|cos(x)|+C}\checkmark$