Question

Нужно найти ! 1. dx/sqrt(x)(x-1) 2. (cos4x - cos6x)/sin5x

Answer 1

$1)\; \int \frac{dx}{ \sqrt{x}(x-1)} =[\, x=t^2\; ,\; t= \sqrt{x}\; ,\; dx=2t\, dt\; ]=\int \frac{2t\, dt}{t\cdot (t^2-1)}=\\\\=\int \frac{2\, dt}{t^2-1}=2\cdot \frac{1}{2}\cdot ln| \frac{t-1}{t+1}|+C=ln| \frac{ \sqrt{x}-1}{ \sqrt{x}+1} |+C$

$2)\int \frac{cos4x-cos6x}{sis5x}dx=\int \frac{-2\cdot sin5x\cdot sin(-x)}{sin5x}dx=2\int sinx\, dx=\\\\=2\cdot (-cosx)+C=-2cosx+C$