Question

Найти неопределенный интеграл

Answer 1

$1)\; \; \int \frac{arccos\, 3x}{\sqrt{1-9x^2}}\, dx=\int \frac{arccos\, 3x}{\sqrt{1-(3x)^2}}\, dx=\Big [\, t=arccos\, 3x,\; dt=-\frac{3dx}{\sqrt{1-(3x)^2}}\, \Big ]=\\\\=-\frac{1}{3}\int t\cdot dt=-\frac{1}{3}\cdot \frac{t^2}{2}+C=-\frac{arccos^23x}{6}+C$

$2)\; \; \int \frac{\sqrt[3]{ctg^2x}}{sin^2x}\, dx=\Big [\, t=ctgx\; ,\; dt=-\frac{dx}{sin^2x}\, \Big ]=-\int t^{2/3}\, dt=\\\\=-\frac{t^{5/3}}{5/3}+C=-\frac{3}{5}\cdot \sqrt[3]{ctg^5x}+C\\\\3)\; \; \int \frac{cosx}{sinx+2}\, dx=\Big [\, t=sinx+2\; ,\; dt=cosx\, dx\, \Big ]=\int \frac{dt}{t}=ln|t|+C=\\\\=ln|sinx+2|+C$