Question

Помогите найти первообразную, сроооочно

Answer 1

$displaystyleint(frac{1}{(3-4x)^7}+frac{2x}{sqrt{6x-3}})dx=-frac{1}{4}intfrac{d(3-4x)}{(3-4x)^7}+\+frac{1}{sqrt{3}}intfrac{2x-1+1}{sqrt{2x-1}}dx=-frac{1}{4}intfrac{d(3-4x)}{(3-4x)^7}+\+frac{1}{2sqrt3}intsqrt{2x-1}d(2x-1)+frac{1}{2sqrt{3}}intfrac{d(2x-1)}{sqrt{2x-1}}=\=frac{1}{24(3-4x)^6}+frac{sqrt{(2x-1)^3}}{3sqrt3}+sqrt{frac{2x-1}{3}}+C$

$displaystyleint (cos^4frac{x}{3}+sin^6frac{x}{3})dx\int cos^4frac{x}{3}dx=frac{1}{4}int(frac{3}{2}+2cosfrac{2x}{3}+frac{1}{2}cosfrac{4x}{3})dx=\=frac{3}{8}int dx+frac{3}{4}int cosfrac{2x}{3}d(frac{2x}{3})+frac{3}{32}int cosfrac{4x}{3}d(frac{4x}{3})=\=frac{3x}{8}+frac{3}{4}sinfrac{2x}{3}+frac{3}{32}sinfrac{4x}{3}+C$
$displaystyle int sin^6frac{x}{3}dx=frac{1}{8}int(1-cosfrac{2x}{3})^3dx=\=frac{1}{8}int(1-3cosfrac{2x}{3}+3cos^2frac{2x}{3}-cos^3frac{2x}{3})dx=\frac{1}{8}int(frac{5}{2}-4cosfrac{2x}{3}+frac{3}{2}cosfrac{4x}{3}+sin^2frac{2x}{3}cosfrac{2x}{3})dx=\=frac{5}{16}x-frac{3}{4}sinfrac{2x}{3}+frac{9}{64}sinfrac{4x}{3}+frac{1}{16}sin^3frac{2x}{3}+C$
$displaystyleint (cos^4frac{x}{3}+sin^6frac{x}{3})dx=frac{11x}{16}+frac{15}{64}sinfrac{4x}{3}+frac{1}{16}sin^3frac{2x}{3}+C$

$displaystyleint(frac{6x^3}{sqrt{6x+4x^2}}+frac{1}{9}sqrt{9x-2})dx\intfrac{6x^3}{sqrt{6x+4x^2}}dx=6intfrac{x^2sqrt{6x+4x^2}}{4x+6}=\=6int(frac{x}{4}-frac{6}{16}+frac{9}{4}frac{1}{4x+6})sqrt{6x+4x^2}dx=\=frac{6}{16}int(4x-6)sqrt{6x+4x^2}dx+frac{27}{2}intfrac{sqrt{x}dx}{sqrt{4x+6}}=\=frac{1}{8}sqrt{(6x+4x^2)^3}-frac{27}{32}((2x+frac{3}{2})sqrt{4x^2+6x}-\-frac{9}{4}ln|2x+frac{3}{2}+sqrt{4x^2+6x}|)+frac{1}{8}(sqrt{4x+6}sqrt{4x}-\-6ln|sqrt{4x+6}+sqrt{4x}|)+C$
$intfrac{1}{9}sqrt{9x-2}dx=frac{1}{81}intsqrt{9x-2}d(9x+2)=frac{2}{243}sqrt{(9x-2)^3}+C$
$displaystyleint(frac{6x^3}{sqrt{6x+4x^2}}+frac{1}{9}sqrt{9x-2})dx=frac{1}{8}sqrt{(6x+4x^2)^3}-\-frac{27}{32}((2x+frac{3}{2})sqrt{4x^2+6x}-frac{9}{4}ln|2x+frac{3}{2}+sqrt{4x^2+6x}|)+\+frac{1}{8}(sqrt{4x+6}sqrt{4x}-6ln|sqrt{4x+6}+sqrt{4x}|)+frac{2}{243}sqrt{(9x-2)^3}+C$

Answer 2

Задания сатанинские... у рядового человека мозг лопнет, пока решит.