Question

Помогите пожалуйста дорешать интегралы.

Answer 1

1)3tg(x/2)|(π/2-π/3)=3(tgπ/4-tgπ/6=3(1-√3/3)=3-√3
2)x√2+3/4*∛(x^4)|(8-0)+8√2+3/4*8*2=8√2+12

Answer 2

$\int _{\frac{\pi}{3}}^{\frac{\pi}{2}}\frac{3dx}{2cos^2\frac{x}{2}}=3tg\frac{x}{2}\, |_{\frac{\pi}{3}}^\frac{\pi}{2}}=3(tg\frac{\pi}{4}-tg\frac{\pi}{6})=3(1-\frac{\sqrt3}{3})=\\\\=3-\sqrt3=\sqrt3(\sqrt3-1)\\\\\\\int _0^8(\sqrt2+\sqrt[3]{x})dx=(\sqrt2x+\frac{x^{\frac{4}{3}}}{\frac{4}{3}})\, |_0^8=(\sqrt2x+\frac{3}{4}\sqrt[3]{x^4})\, |_0^8=\\\\=\sqrt2\cdot 8+\frac{3}{4}\sqrt[3]{8^4}=8\sqrt2+\frac{3}{4}\cdot 2^4=8\sqrt2+12=4(2\sqrt2+3)$