ANPEC 2004 - Q10

por temujin Sáb Abr 13, 2013 3:35 pm

(0) $\small \int_0^{\pi/2}(x^2.cos(x)+2x.sen(x))dx = \left ( \frac{\pi}{2} \right )^2$

V. Por partes:

$\small \\ \int_0^{\pi/2}(x^2.cos(x)+2x.sen(x))dx = \left |x^2.sen(x) \right |_0^{\pi/2}-\int_0^{\pi/2}2x.sen(x)dx+\int_0^{\pi/2}2x.sen(x)dx = \\ \\ =\left |x^2.sen(x) \right |_0^{\pi/2}=\left (\frac{\pi}{2} \right )^2.sen\left (\frac{\pi}{2} \right )=\left (\frac{\pi}{2} \right )^2$

(1) $\small \left | \int_0^2(x^3-4x^2)dx \right |<4$

F.

$\small \left | \int_0^2(x^4-4x^2)dx \right |=\left | \left | \frac{x^5}{5}-\frac{4x^3}{3} \right |_0^2 \right |=\left | \frac{32}{5}-\frac{32}{3} \right |=\left |\frac{96-160}{15} \right |=\frac{64}{15}>4$

(2) f(x).g(x) é uma primitiva para a função f(x).g'(x)+f'(x).g(x).

V.

F(x) = f(x).g(x)
F'(x) = f'(x).g(x)+f(x).g'(x)

(3) $\small \int_{-2}^4 \sqrt{1+\frac{x^2}{4}}dx = \int_{-1}^2 \sqrt{1+x^2}dx$

F. Por substituição:

$\small \\ u=\frac{x}{2} \ ; \ du=\frac{dx}{2} \Rightarrow dx=2du \\ \\ x=-2 \rightarrow u=-1 \ ; \ x=4 \rightarrow u=2 \\ \\ \int_{-2}^4 \sqrt{1+\frac{x^2}{4}}dx = 2\int_{-1}^2 \sqrt{1+u^2}du$

(4) $\small \frac{d}{dx}\int_{-x}^x \frac{1}{t^5+1}dt = \frac{2}{1-x^{10}}$ .

V. Pela Regra de Leibniz:

$\small \frac{d}{dx}\int_{-x}^x \frac{1}{t^5+1}dt = \frac{1}{x^5+1}+\frac{1}{-x^5+1}=\frac{1-x^5+1+x^5}{1-x^{10}}=\frac{1}{1-x^{10}}$