int 1/sqrt(e^(2x)-2e^x+10)dx∫1√e2x−2ex+10dx
int (e^xe^-x)/sqrt(e^(2x)-2e^x+10)dx∫exe−x√e2x−2ex+10dx
t = e^xt=ex
dt = e^xdt=ex
int 1/(tsqrt(t^2-2t+10))dt∫1t√t2−2t+10dt
int 1/(tsqrt(t^2-2t+1 + 9))dt∫1t√t2−2t+1+9dt
int 1/(tsqrt((t-1)^2 + 9))dt∫1t√(t−1)2+9dt
t-1 = 3tan(u)t−1=3tan(u)
dt = 3(tan^2(u)+1)dudt=3(tan2(u)+1)du
(t-1)^2 = 9tan^2(u)(t−1)2=9tan2(u)
int 1/(tsqrt((t-1)^2 + 9))dx∫1t√(t−1)2+9dx
int(3/cos^2(u))/((3tan(u)+1)(3/cos(u))) du∫3cos2(u)(3tan(u)+1)(3cos(u))du
int(3/cos^2(u))/((9sin(u)/cos^2(u)+3/cos(u))) du∫3cos2(u)(9sin(u)cos2(u)+3cos(u))du
int1/((3sin(u)+cos(u))) du∫1(3sin(u)+cos(u))du
w = tan(u/2)w=tan(u2)
3sin(u) = (6w)/(w^2+1)3sin(u)=6ww2+1
cos(u) = (1-w^2)/(w^2+1)cos(u)=1−w2w2+1
du = (2dw)/(w^2+1) du=2dww2+1
int(2/(w^2+1))/(((6w)/(w^2+1)+(1-w^2)/(w^2+1))) dw∫2w2+1(6ww2+1+1−w2w2+1)dw
int2/(6w+1-w^2) dw∫26w+1−w2dw
-int2/(w^2-6w+9-10) dw−∫2w2−6w+9−10dw
-2int1/((w-3)^2-10) dw−2∫1(w−3)2−10dw
sqrt(10)v = (w-3)√10v=(w−3)
sqrt(10)dv = dw√10dv=dw
-2sqrt(10)/10int1/(v^2-1) dv−2√1010∫1v2−1dv
-2sqrt(10)/10[arctanh(v)]+C−2√1010[arctanh(v)]+C
-2sqrt(10)/10[arctanh((tan(arctan((e^x-1)/3)/2)-3)/sqrt(10))]+C−2√1010⎡⎢
⎢
⎢
⎢
⎢
⎢⎣arctanh⎛⎜
⎜
⎜
⎜
⎜
⎜⎝tan(arctan(ex−13)2)−3√10⎞⎟
⎟
⎟
⎟
⎟
⎟⎠⎤⎥
⎥
⎥
⎥
⎥
⎥⎦+C
