Вычислите следующие интегралы .

Решите задачу:

$\int _0^1e^{3x}dx=\frac{1}{3}e^{3x}\, |_0^1=\frac{1}{3}(e^3-1)\\\\\int _2^3\frac{dx}{x-1}=ln|x-1|\, |_2^3=ln2-ln1=ln2\\\\\int _1^2\frac{dx}{x+3}=ln|x+3|\, |_1^2=ln5-ln4=ln\frac{5}{4}$

$\int _{-\frac{\pi}{4}}^{\frac{\pi}{4}}cosx\, dx=sinx\, |_{-\frac{\pi}{4}}^{\frac{\pi}{4}}=sin\frac{\pi}{4}-sin(-\frac{\pi}{4})=\frac{\sqrt2}{2}+\frac{\sqrt2}{2}=\sqrt2\\\\\int _{-\frac{\pi}{2}}^{\frac{\pi}{2}}(cosx-sinx)dx=(sinx+cosx)|_{-\frac{\pi}{2}}^{\frac{\pi}{2}}=\\\\=sin\frac{\pi}{2}+cos\frac{\pi}{2}-(sin(-\frac{\pi}{2})+cos(-\frac{\pi}{2}))=1+0-(-1+0)=2$