Question

Вычислите интеграл: 1) ∫от 0 до π/2 (2sin2x-1/3cos x/3) dx ; 2) ∫ от 0 до 6 (x+(5/ √ 0.5x+1) -x) dx

Answer 1

$1)\; \; \int\limits^{\frac{\pi}{2}}_0(2sin2x-\frac{1}{3}cos \frac{x}{3}) \, dx=(-2\cdot \frac{1}{2}cos2x-\frac{1}{3}\cdot 3\cdot sin\frac{x}{3})\Big |_0^{\frac{\pi}{2}}=\\\\=-cos\pi -sin\frac{\pi}{6}-(-cos0-sin0)=-(-1)-\frac{1}{2}+1+0=1\frac{1}{2}\\\\\\2)\; \; \int\limits^6_0(x+ \frac{5}{\sqrt{0,5x+1}}-x)dx= \int\limits^6_0 \frac{5}{\sqrt{0,5x+1}} \, dx=5\cdot \frac{1}{0,5}\cdot 2\sqrt{0,5x+1}\Big |_0^6=\\\\=20\cdot (\sqrt{4}-\sqrt1)=20\cdot (2-1)=20\\\\\\\star \int sin(ax+b)dx=-\frac{1}{a}cos(ax+b)+C$

$\star \; \; \int cos(ax+b)dx=\frac{1}{a}sin(ax+b)+C\\\\\star \; \; \int \frac{dx}{\sqrt{ax+b}}= \frac{1}{a}\cdot 2\sqrt{ax+b}+C$