Substitution?

[Ryan Rigdon]

Having trouble with this problem. Can someone help me figure out how they went from the left integral to the right integral.

 

[lookagain]

Having trouble with this problem. Can someone help me figure out how they went from the left integral to the right integral.

If \(\displaystyle x = 3\tan t\), then

\(\displaystyle dx = 3(\sec^2 t) dt\), which is the newer numerator.

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Working out the newer denominator:

\(\displaystyle (x^2 + \ 9)^2 = ((3\tan t)^2 + \ 9)^2 = (9\tan^2 t + \ 9)^2 = [9(\tan^2 t+ \ 1)]^2\)


Note the identity: \(\displaystyle \tan^2 t + 1 = \sec^2 t\)


Continuing from two lines above:

\(\displaystyle [9(\tan^2t + \ 1)]^2 = 81(\sec^2t)^2 = 81\sec^4t\)

 

[Ryan Rigdon]

hey thanx bro. now it makes sense. this will help me solve the rest of the problem now. thanx again. peace