Question

Could someone please verify that I solved the following integral correctly (click to see).

Answer 1

\[\int\limits_{}^{}\tan(x)\sec ^{3}(x)dx\]
\[=\int\limits_{}^{}\tan(x)\sec(x)\sec ^{2}(x)dx\]
\[=\int\limits_{}^{}\tan(x)\sec(x)u ^{2}\frac{ du }{ \sec(x)\tan(x) }\]
\[=\int\limits_{}^{}u ^{2}du=\frac{ \sec ^{3}(x) }{ 3 }+C\]

Answer 2

Idk if what I did was legal, but I got the right answer

Answer 3

Wes you solved it correctly, though you mixed your variables. You should never have u and x in the same expression.

Answer 4

Yes*

Answer 5

I've seen u's and x's in the same expression many times, that's how x values are canceled and u's remain

Answer 6

I would have written\[\int\sec^3x\tan xdx=\]\[\int\sec^2x\sec x\tan xdx\]\[u=\sec x\implies du=\sec x\tan xdx\]and just subed in
i'm not saying that what you did is totally wrong, I'm just saying it's a bad habit, and may confuse you later on. I do the trick you are referring to if I have only a constant that appears due to the u-sub, but with other variables it can get confusing.
Still, right answer, nice job.