Question

Sabendo-se que a integral indefinida de ∫ cos x dx = sen x+c e que é válida a identidade trigonométrica sen²x + cos²x = 1 , o valor da integral indefinida abaixo indicada é:
∫ cos³x dx

Answer 1

$\boxed{\begin{array}{l}\displaystyle\sf\int cos^3(x)~dx=\int cos^2(x)~cos(x)\,dx\\\displaystyle\sf\int cos^3(x)~dx=\int(1-sen^2(x)) cos(x)~dx\\\underline{\rm fac_{\!\!,}a} \\\sf t=sen(x)\implies dt=cos(x)~dx\\\displaystyle\sf\int cos^3(x)=\int(1-t^2)dt=t-\dfrac{1}{3}t^3+c\\\sf como~t=sen(x)~ent\tilde ao\\\displaystyle\sf\int cos^3(x)~dx=sen(x)-\dfrac{1}{3}sen^3(x)+c\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~a}}}}\end{array}}$

$\displaystyle\sf\ell ife=\int_{happy}^{death}\dfrac{happines}{time}d_{time}$