Question

Найти неопределенные интегралы

Answer 1

$LARGE \ 1)int{e^{3x}over sqrt{e^{6x}-7}}mathrm{dx}=begin{vmatrix} e^{3x}=t\ mathrm{dx}={mathrm{dt}over3t} end{vmatrix}={1over3}int{tmathrm{dt}over tsqrt{t^2-7}}={1over3}ln{|e^{3x}+sqrt{e^{6x}-7}|}+C\ 2)int{sin{2x}over3+sin^2{x}}mathrm{dx}=2int{sin{x}cos{x}over3+sin^2{x}}mathrm{dx}=begin{vmatrix} sin{x}=t\ mathrm{dx}={mathrm{dt}over cos{x}} end{vmatrix}=2int {tmathrm{dt}over3+t^2}=int{mathrm{d(3+t^2)}over3+t^2}=ln{|3+t^2|}+C=ln{|3+sin^2{x}|}+C\ 3)int(2x-1)sin{3x}mathrm{dx}=begin{vmatrix} u=2x-1,mathrm{du}=2mathrm{dx}\ mathrm{dv}=sin{3x}mathrm{dx}, v=-{cos{3x}over3} end{vmatrix}= -{(2x-1)cos{3x}over3}-{-2over3}intcos{3x}mathrm{dx}={2over9}sin{3x}-{(2x-1)cos{3x}over3}+C$