Question

O valor da integral cos^2(x)senx de 3pi/2 ate 0

Answer 1

$\displaystyle \int\limits^{0}_{\frac{3\pi}{2}}\cos^2(x)\cdot \sin(x)\,dx=-\int\limits_{0}^{\frac{3\pi}{2}}\cos^2(x)\cdot\sin(x)\,dx\implies \\\\\cos(x)=u\implies \frac{du}{dx}=\sin(x)\implies du=-\sin(x)\cdot dx\implies \\\\-\int\limits_{0}^{\frac{3\pi}{2}}-u^2\,du=\int\limits^{\frac{3\pi}{2}}_{0}u^2\,du=\left.\frac{u^3}{3}\right|\limits_{0}^{\frac{3\pi}{2}}=\left.\frac{\cos^3(x)}{3}\right|\limits_{0}^{\frac{3\pi}{2}}=\\\\\frac{\cos^3(\frac{3\pi}{2})}{3}-\frac{\cos^3(0)}{3}=\frac{\cos^3(\frac{3\pi}{2})-\cos^3(0)}{3}=\frac{0^3-1^3}{3}=\frac{-1}{3}=\boxed{-\frac{1}{3}}$