Сначала раскрыть модуль
65m+70+45m-24
Потом привести подобные члены и вычесть.
110m+46
![\int \frac{\sqrt{1+x}}{x}dx](https://tex.z-dn.net/?f=%5Cint%20%20%5Cfrac%7B%5Csqrt%7B1%2Bx%7D%7D%7Bx%7Ddx%20)
Замена:
![1+x=t^2\\ x=t^2-1\\ dx=2tdt](https://tex.z-dn.net/?f=1%2Bx%3Dt%5E2%5C%5C%0Ax%3Dt%5E2-1%5C%5C%0Adx%3D2tdt)
![\int \frac{ \sqrt{t^2}}{t^2-1}\cdot 2tdt=2\int \frac{t^2dt}{t^2-1}=2 \int \frac{(t^2-1)+1}{t^2-1}dt=2\int[ \frac{t^2-1}{t^2-1}+ \frac{1}{t^2-1}]dt=\\\\ 2\int[1+ \frac{1}{t^2-1}]dt=2\int dt+2\int \frac{1}{t^2-1}dt=2t+2\int \frac{1}{(t-1)(t+1)}dt=2t+2Y](https://tex.z-dn.net/?f=%5Cint%20%20%5Cfrac%7B%20%5Csqrt%7Bt%5E2%7D%7D%7Bt%5E2-1%7D%5Ccdot%202tdt%3D2%5Cint%20%20%5Cfrac%7Bt%5E2dt%7D%7Bt%5E2-1%7D%3D2%20%5Cint%20%5Cfrac%7B%28t%5E2-1%29%2B1%7D%7Bt%5E2-1%7Ddt%3D2%5Cint%5B%20%5Cfrac%7Bt%5E2-1%7D%7Bt%5E2-1%7D%2B%20%5Cfrac%7B1%7D%7Bt%5E2-1%7D%5Ddt%3D%5C%5C%5C%5C%0A2%5Cint%5B1%2B%20%5Cfrac%7B1%7D%7Bt%5E2-1%7D%5Ddt%3D2%5Cint%20dt%2B2%5Cint%20%5Cfrac%7B1%7D%7Bt%5E2-1%7Ddt%3D2t%2B2%5Cint%20%5Cfrac%7B1%7D%7B%28t-1%29%28t%2B1%29%7Ddt%3D2t%2B2Y)
,
где
![Y=\int \frac{1}{(t-1)(t+1)}dt](https://tex.z-dn.net/?f=Y%3D%5Cint%20%5Cfrac%7B1%7D%7B%28t-1%29%28t%2B1%29%7Ddt%20)
решим разложением на две простые дроби
![\frac{1}{(t-1)(t+1)}= \frac{a}{t-1}+ \frac{b}{t+1}= \frac{a(t+1)+b(t-1)}{(t-1)(t+1)}= \frac{at+a+bt-b}{(t-1)(t+1)}= \frac{(a+b)t+(a-b)}{(t-1)(t+1)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B%28t-1%29%28t%2B1%29%7D%3D%20%5Cfrac%7Ba%7D%7Bt-1%7D%2B%20%5Cfrac%7Bb%7D%7Bt%2B1%7D%3D%20%5Cfrac%7Ba%28t%2B1%29%2Bb%28t-1%29%7D%7B%28t-1%29%28t%2B1%29%7D%3D%20%5Cfrac%7Bat%2Ba%2Bbt-b%7D%7B%28t-1%29%28t%2B1%29%7D%3D%20%5Cfrac%7B%28a%2Bb%29t%2B%28a-b%29%7D%7B%28t-1%29%28t%2B1%29%7D)
![\left \{ {{a+b=0} \atop {a-b=1}} \right. \Longrightarrow 2a=1 \Longrightarrow a= \frac{1}{2}; b=- \frac{1}{2}](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7Ba%2Bb%3D0%7D%20%5Catop%20%7Ba-b%3D1%7D%7D%20%5Cright.%20%5CLongrightarrow%202a%3D1%20%5CLongrightarrow%20a%3D%20%5Cfrac%7B1%7D%7B2%7D%3B%20b%3D-%20%5Cfrac%7B1%7D%7B2%7D%20)
Тогда
![\int \frac{1}{(t-1)(t+1)}dt= \frac{1}{2}\int \frac{1}{t-1}dt- \frac{1}{2}\int \frac{1}{(t+1)}dt= \\\\ \frac{1}{2}\int \frac{1}{(t-1)}d(t-1)- \frac{1}{2}\int \frac{1}{(t+1)}d(t+1)= \frac{1}{2}\ln|t-1|- \frac{1}{2}\ln |t+1|=\\\\ \frac{1}{2}(\ln |t-1|-\ln|t+1|)= \frac{1}{2}\ln| \frac{t-1}{t+1}|= \frac{1}{2}\ln | \frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}|](https://tex.z-dn.net/?f=%5Cint%20%20%5Cfrac%7B1%7D%7B%28t-1%29%28t%2B1%29%7Ddt%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cint%20%20%5Cfrac%7B1%7D%7Bt-1%7Ddt-%20%5Cfrac%7B1%7D%7B2%7D%5Cint%20%20%5Cfrac%7B1%7D%7B%28t%2B1%29%7Ddt%3D%20%5C%5C%5C%5C%20%0A%5Cfrac%7B1%7D%7B2%7D%5Cint%20%20%5Cfrac%7B1%7D%7B%28t-1%29%7Dd%28t-1%29-%20%5Cfrac%7B1%7D%7B2%7D%5Cint%20%5Cfrac%7B1%7D%7B%28t%2B1%29%7Dd%28t%2B1%29%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cln%7Ct-1%7C-%20%5Cfrac%7B1%7D%7B2%7D%5Cln%20%7Ct%2B1%7C%3D%5C%5C%5C%5C%0A%20%5Cfrac%7B1%7D%7B2%7D%28%5Cln%20%7Ct-1%7C-%5Cln%7Ct%2B1%7C%29%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cln%7C%20%5Cfrac%7Bt-1%7D%7Bt%2B1%7D%7C%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cln%20%7C%20%5Cfrac%7B%5Csqrt%7B1%2Bx%7D-1%7D%7B%5Csqrt%7B1%2Bx%7D%2B1%7D%7C%20%20)
Тогда ответ:
![2\sqrt{1+x}+2\cdot \frac{1}{2}\ln |\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}|+C=\\\\ 2\sqrt{1+x}+\ln | \frac{\sqrt{1+x}-1}{\sqrt{1+x}+1} |+C](https://tex.z-dn.net/?f=2%5Csqrt%7B1%2Bx%7D%2B2%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Cln%20%7C%5Cfrac%7B%5Csqrt%7B1%2Bx%7D-1%7D%7B%5Csqrt%7B1%2Bx%7D%2B1%7D%7C%2BC%3D%5C%5C%5C%5C%202%5Csqrt%7B1%2Bx%7D%2B%5Cln%20%7C%20%5Cfrac%7B%5Csqrt%7B1%2Bx%7D-1%7D%7B%5Csqrt%7B1%2Bx%7D%2B1%7D%20%7C%2BC)
, где C- константа
Раз внешний угол=40 ,то угол при вершине=140 .
следовательно,сумма угол в треугольнике равна 180,то мы 180-140=40.
и т.к у нас треугольник равнобедренный то 40/2=20