Question

Triq substitution integration question:

Answer 1

\[\int\limits_{\sqrt{2}}^{2} {\frac{ 1 }{ t^3\sqrt{t^2-1} }}dt\]

Answer 2

i subbed t = sec(a), so dt=da*sec(a)*tan(a)

Answer 3

so i come out with \[\int\limits_{}^{}\cos^2(a)da\] but don't know where to go from there

Answer 4

double-angle formula

Answer 5

\[\cos^2\theta=\frac12[1+\cos(2\theta)]\]

Answer 6

\[\frac{ 1 }{ 2 } \int\limits_{}^{}da +\frac{ 1 }{ 2 } \int\limits_{}^{}\cos(2a)da\]

Answer 7

yep

Answer 8

but because t=sec(a), a=arcsec(t) and then the answer gets super complicated

Answer 9

I would change the bounds to be in terms of theta to make this integral easier to evaluate
you don't need to sub back in for x

Answer 10

\[t=\sec u\]\[\sqrt 2=\sec u\implies u=\frac\pi4\]\[2=\sec u\implies u=\frac\pi3\]

Answer 11

now just do the integral with those bounds...

Answer 12

thanks for the help :) got the answer right :)

Answer 13

pi/24 + sqrt(3)/8 - 1/4