Question

integral tan^5 4x dx

Answer 1

Hey Biz :)

Hmmmm an odd power of tangent. Thinkinggg......

\[\Large\rm \int\limits \tan^54x~dx=\int\limits (\tan4x)^2 \tan^34x~dx\]\[\Large\rm \]\[\Large\rm \int\limits (\sec^24x-1)\tan^34x~dx\]\[\Large\rm =\int\limits \tan^34x~\sec^24x~dx-\int\limits \tan^34x~dx\]In the first integral you can make the substitution: \(\Large\rm u=tan4x\)
And it should work out nicely since your du will involve a sec^2(4x).

Answer 2

For the second integral, you'll have to change some more tangents into secants D:

Answer 3

\[\Large\rm \int\limits \tan^34x~dx=\int\limits (\sec^24x-1)\tan 4x~dx\]\[\Large\rm =\int\limits \tan4x~\sec^24x~dx-\int\limits \tan4x~dx\]

Answer 4

We get another integral that's easy to u-sub.
And then you'll want to recall your tangent integral.
If you don't remember it, you can write it as sin4x/cos4x and do u=cos4x.

Answer 5

Thnx :)