Question

Indefinite integral problem with trig functions using substitution:

integral of 5 * sec^2(5x) * tan^5(5x) dx

Answer 1

u = tan(5x), du = 5(sec(5x))^2 dx

Answer 2

\(\large\color{slate}{\displaystyle 5\int\limits_{~}^{~}\sec^2(5x)\times \tan^5(5x)~dx}\)

this?

Answer 3

then yes, u=tan(5x) as was suggested, because the derivative of tan(5x) is 5sec^2(5x) (sitting inside there)

Answer 4

yes though the five is inside the integral, though that shouldn't matter

Answer 5

yes, so your derivative of "u" is sitting inside the integral for you already

Answer 6

I see how that works. What am I left with after I use that tan(5x)? tan^4(5x)?

Answer 7

no

Answer 8

u^5

Answer 9

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}5\sec^2x\tan^5(5x)~dx}\)

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}5(\sec~x)^2~(\tan~5x)^5~dx}\)


\(\large\color{blue}{\displaystyle u=\tan ~5x}\)
\(\large\color{green}{\displaystyle du=5(\sec ~5x)^2~dx}\)


\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\color{green}{5(\sec~x)^2}~(\color{blue}{\tan~5x})^5~\color{green}{dx} }\)

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}(\color{blue}{u})^5~\color{green}{du} }\)

Answer 10

and the \(\large\color{green}{ 5(\sec~5x)^2 }\) went away with \(\large\color{green}{ dx }\) TO BE REPLACED BY \(\large\color{green}{ du }\)

Answer 11

Oh Okay that makes sense! I always forget the laws for trig functions. Okay can I finish this and see if we got the same answer?

Answer 12

yes, apply the power rule, and then sub the tan(5x) back for "u" as you are done.

Answer 13

So it should be 1/6 * tan^6(5x)?

Answer 14

yes, with one more thing.... (you left something out)

Answer 15

+ ... ?

Answer 16

+ C

Answer 17

Yup

Answer 18

uggggggghhhhh I always forget that one after a long problem

Answer 19

\(\large\color{black}{ \displaystyle \frac{\tan^6(5x)}{6} ~+C }\)

Answer 20

I would take almost all points off for leaving out +C

Answer 21

because when you don't write +C you only include 1 possible answer out of the infinite number of answers there are.

Answer 22

Good integration, but +C is very important, because not only \(\large\color{black}{ \displaystyle \frac{\tan^6(5x)}{6} ~ }\), but also ANY \(\large\color{black}{ \displaystyle \frac{\tan^6(5x)}{6} ~+C }\) will have a slope of \(\large\color{black}{ \displaystyle 5\sec^2(5x)\tan^5(5x) }\).

Answer 23

(good job though...) yw

Answer 24

Thanks Solomon. Your name fits you!

Answer 25

You mean it as the King Solomon? Tnx for the complement, I will take it:D

Answer 26

absolutely!

Answer 27

You too will surely get there... you are moving very quickly in your math latter.