Question

Решите 1,2,6 задания

Answer 1

$1)\; \; \int (5x^9-\frac{3}{x}+6\, sin2x)dx=5\cdot \frac{x^{10}}{10}-3\cdot ln|x|-6\cdot \frac{1}{2}\cdot cos2x+C\\\\2)\; \; \int (8x+9)\, cos6x\, dx=[\; u=8x+9\; ,du=8\, dx\; ,\; \; dv=cos6x\, dx\; ,\\\\v=\frac{1}{6}\, sin6x\; ]=uv-\int v\, du=\frac{1}{6}(8x+9)sin\, 6x-\frac{8}{6}\int sin6x\, dx=\\\\=\frac{1}{6}(8x+9)sin6x+\frac{8}{36}\cdot cos6x+C=\frac{1}{6}(8x+9)sin6x+\frac{2}{9}\cdot cos6x+C$

$6)\; \; \int cos^{-6}x\cdot sinx\, dx=\int \frac{sinx}{cos^6x}\, dx=[\; t=cosx\; ,\; dt=-sinx\, dx\; ]=\\\\=-\int \frac{dt}{t^6}=-\int t^{-6}\, dt=-\frac{t^{-5}}{-5}=\frac{1}{5\, cos^5x}+C$