How do you integrate dx(9x225)32 using trig substitutions?

1 Answer
Nov 24, 2016

The answer is =x259x225+C

Explanation:

We use sec2u=1+tan2u

Let's use the substitution

x=5secu3 , dx=5secutanudu3

dx(9x225)32=5secutanudu3(9259sec2u25)32

=5secutanudu3125(tan3u)

=175secudutan2u

=175cos2uducosusin2u

=175cosudusin2u

Let v=sinu,dv=cosudu

Therefore,

175cosudusin2u=175dvv2

=175v2dv=175v11=175v

=175sinu

x=5secu3 , cosu=53x

sin2u=1cos2u=1259x2=9x2259x2

1sinu=3x9x225

Therefore,

dx(9x225)32=(175)3x9x225+C

=x259x225+C