Is it possible to solve this integrate by integrate substitution method?

int tanx/secx

Not this away: int [(senx)/(cosx)]cosx/1

int senx

-cos x + c

2 Answers
Jim H
Apr 2, 2018

Here is one way. (I think it is harder than what you did.)

Explanation:

int tanx/secx dx

Let u = tanx. Then du = sec^2x dx and

secx = sqrt(tan^2x+1) = sqrt(u^2+1) So that du = (u^2+1) dx and dx = 1/(u^2+1) du

int tanx/secx dx = int u/sqrt(u^2+1) * 1/(u^2 +1) du

= int u (u^2+1)^(-3/2) du

= 1/2 int 2u (u^2+1)^(-3/2) du

= 1/2[-2/1 (u^2+1)^(-1/2)]+C

= -1/(u^2+1)^(1/2)+C

= -1/(u^2+1)^(1/2)+C

= -1/sqrt (sec^2x)+C

= -1/secx+C

= -cosx+C

Jim H
Apr 3, 2018

Here is another substitution that will work.

Explanation:

Let u = secx so du = secxtanxdx and

int tanx/secx dx = int 1/sec^2x secx tanx dx

= int 1/u^2 du

= -u^(-1)+C = -1/u +C

= -1/secx +C

= -cosx +C

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