Пусть y = uv, тогда y' = u'v + uv':
Решим левый интеграл:
cosx = \frac{1-t^2}{1+t^2} => dx = \frac{2}{1+t^2}dt\\ \int \frac{2(1+t^2)}{(1+t^2)(1-t^2)} dt = \int \frac{2}{(1-t)(1+t)}dt = \int ( \frac{1}{1-t} + \frac{1}{1+t})dt = ln(1-t)+ln( 1+t) = ln|1-t^2| = ln|1-tg^2\frac{x}{2}| \\" class="latex-formula" id="TexFormula2" src="https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7Bdx%7D%7Bcosx%7D%3B%5C%5C%20tg%5Cfrac%7Bx%7D%7B2%7D%3Dt%20%3D%3E%20cosx%20%3D%20%5Cfrac%7B1-t%5E2%7D%7B1%2Bt%5E2%7D%20%3D%3E%20dx%20%3D%20%5Cfrac%7B2%7D%7B1%2Bt%5E2%7Ddt%5C%5C%20%20%5Cint%20%5Cfrac%7B2%281%2Bt%5E2%29%7D%7B%281%2Bt%5E2%29%281-t%5E2%29%7D%20dt%20%3D%20%5Cint%20%5Cfrac%7B2%7D%7B%281-t%29%281%2Bt%29%7Ddt%20%3D%20%5Cint%20%28%20%5Cfrac%7B1%7D%7B1-t%7D%20%2B%20%5Cfrac%7B1%7D%7B1%2Bt%7D%29dt%20%3D%20ln%281-t%29%2Bln%28%201%2Bt%29%20%3D%20ln%7C1-t%5E2%7C%20%3D%20ln%7C1-tg%5E2%5Cfrac%7Bx%7D%7B2%7D%7C%20%20%5C%5C" title="\int \frac{dx}{cosx};\\ tg\frac{x}{2}=t => cosx = \frac{1-t^2}{1+t^2} => dx = \frac{2}{1+t^2}dt\\ \int \frac{2(1+t^2)}{(1+t^2)(1-t^2)} dt = \int \frac{2}{(1-t)(1+t)}dt = \int ( \frac{1}{1-t} + \frac{1}{1+t})dt = ln(1-t)+ln( 1+t) = ln|1-t^2| = ln|1-tg^2\frac{x}{2}| \\">
Возвращаемся к исходному:
Пусть y = uv, тогда y' = u'v + uv':
Решим левый интеграл:
cosx = \frac{1-t^2}{1+t^2} => dx = \frac{2}{1+t^2}dt\\ \int \frac{2(1+t^2)}{(1+t^2)(1-t^2)} dt = \int \frac{2}{(1-t)(1+t)}dt = \int ( \frac{1}{1-t} + \frac{1}{1+t})dt = ln(1-t)+ln( 1+t) = ln|1-t^2| = ln|1-tg^2\frac{x}{2}| \\" class="latex-formula" id="TexFormula2" src="https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7Bdx%7D%7Bcosx%7D%3B%5C%5C%20tg%5Cfrac%7Bx%7D%7B2%7D%3Dt%20%3D%3E%20cosx%20%3D%20%5Cfrac%7B1-t%5E2%7D%7B1%2Bt%5E2%7D%20%3D%3E%20dx%20%3D%20%5Cfrac%7B2%7D%7B1%2Bt%5E2%7Ddt%5C%5C%20%20%5Cint%20%5Cfrac%7B2%281%2Bt%5E2%29%7D%7B%281%2Bt%5E2%29%281-t%5E2%29%7D%20dt%20%3D%20%5Cint%20%5Cfrac%7B2%7D%7B%281-t%29%281%2Bt%29%7Ddt%20%3D%20%5Cint%20%28%20%5Cfrac%7B1%7D%7B1-t%7D%20%2B%20%5Cfrac%7B1%7D%7B1%2Bt%7D%29dt%20%3D%20ln%281-t%29%2Bln%28%201%2Bt%29%20%3D%20ln%7C1-t%5E2%7C%20%3D%20ln%7C1-tg%5E2%5Cfrac%7Bx%7D%7B2%7D%7C%20%20%5C%5C" title="\int \frac{dx}{cosx};\\ tg\frac{x}{2}=t => cosx = \frac{1-t^2}{1+t^2} => dx = \frac{2}{1+t^2}dt\\ \int \frac{2(1+t^2)}{(1+t^2)(1-t^2)} dt = \int \frac{2}{(1-t)(1+t)}dt = \int ( \frac{1}{1-t} + \frac{1}{1+t})dt = ln(1-t)+ln( 1+t) = ln|1-t^2| = ln|1-tg^2\frac{x}{2}| \\">
Возвращаемся к исходному:
2) -7y+5+4,5y-1= -2.5y+4=-5/2*4/5+4=-2+4=2
3) а) x=36
б) 6x=10.2
x=1.7
в) 5x-3x=2,5+4,5
2x=7
x=3.4=5
г) 2x-6x+5=45
-4x=40
x=-10
4) x+x+6=26
2x=20
x=10 минут едет на автобусе
5) 3x-20=x+10
2x=30
x=15т во втором сарае
15*3=45т в первом сарае
6) 7x-x-3=6x-3
6x-3=6x-3
1≡1
7) 14-132/10:(74/21-34/15)=14-66/5:(74*5/105-34*7/105)= 14-66/5:(370/105-238/105)=14-66/5:132/105= 14-66/5*105/132= 14- 21/2= 7/2=3.5
8) х+0.42х+0.28х=320
1.7x=320 (скорее всего опечатка и там 340 страниц).
если 340, то
x=200 страниц первая глава
200*0.42=84 страницы вторая глава
200*0.28=56 страниц третья глава
9) 5/12y+13/10=53/100+7/8y
5y/12-7y/8=53/100-13/10
5*2y/24- 7*3y/24= 53/100-130/100
10y/24-21y/24= -77/100
-11y/24=-77/100
11*100y=77*24
1100y=1848
y=1848/1100
y=1.68
10) 11/6:22/3=16/10:x
11/6*3/22=8/5*1/x
1/4=8/5x
1*5x=8*4
5x=32
x=6.4
11) 40:100*80=32
4/7n=32
4n=32*7
n=32*7:4
n=56
12) a:a/b= a*b/a= b
13) -3/5*-5/9*m/2*-9/5= 1/3*m/2*-9/5= -9m/3*2*5= - 3m/10= -0.3m
14) -6/25:(17/6:34/15-3/2*3/10)= -6/25:( 17/6*15/34-9/20)= -6/25:(15/12-9/20)=-6/25:(15*5/60-9*3/60)=-6/25:(75/60-27/60)= -6/25:48/60= -6/25*60/48= - 3/10= -0.3