Question

Encontre a integral indefinida para ∫[sin(x)]^3dx

A) (([cos(x)]^3)/3)+C
B) −cos(x)+(([cos(x)]^3)/3)+C
C) −sin(x)+(([cos(x)]^2)/3)+C
D) −cos(x)+C
E) −sin(x)+(([cos(x)]^2)/4)+C

Answer 1

$\displaystyle \sf \int [sin(x)]^3 dx \\\\\\ \int sin(x)\cdot sin^2(x)dx \\\\\\ \int sin(x)\cdot (1-cos^2(x)) dx \\\\\\ Fa{\c c}amos : \\\\ cos(x)=u \to sin(x)dx=-du \\\\ Da{\'i}} : \\\\ \int -du(1-u^2)du \\\\ \int u^2du -\int du =\frac{u^3}{3} -u+C \\\\\\ \huge\boxed{\sf -cos(x)+\frac{cos^3(x)}{3}+C\ }\checkmark$

item B