Question

Resolva a integral por substituição

Answer 1

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$\displaystyle\sf\int_0^{\frac{\pi}{2}}sen(x)~cos^2(x)~dx\\\underline{\rm fac_{\!\!,}a}~ t=cos(x)\implies -dt=sen(x)dx\\\sf se~x=0\implies u=1\\\sf se~x=\dfrac{\pi}{2}\implies u=0\\\displaystyle\sf\int_0^{\frac{\pi}{2}} sen(x)\cdot cos^2(x)~dx=-\int_1^0 t^2~dt=\int_0^1 t^2~dt=\bigg[\dfrac{1}{3}t^3\bigg]_0^1\\\sf=\dfrac{1}{3}\cdot1^3-\dfrac{1}{3}\cdot0=\dfrac{1}{3}\\\huge\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int_0^{\frac{\pi}{2}}s en(x)\cdot cos^2(x)~dx=\dfrac{1}{3}}}}}\blue{\checkmark}$