Integrate tan(x)*Sec(x)?

Question

How do we integrate tan(x)*Sec(x). Would it be by substitution, letting u = tanx?

Answer 1

Let u = tan x
du = sec x dx
I = ∫ u du
I = u ² / 2 + C
I = tan ² x + C

Answer 2

Integrate it directly.

∫[(tan x)(sec x)] dx = sec x + C

You need to know the derivatives of the basic trig functions.

Answer 3

you could try u = sec x

du = sec x tan x dx

so that the integral becomes integral dt = t + C = sec x +C

Answer 4

int(tanx * secx)dx = int(sinx / (cosx)^2)dx

Substitute u = cosx
du = -sinx dx

int(-u^-2)du = u^-1 + C

But u = cosx
So:
u^-1 + C = secx + C

Remember your trig identities when integrating! There are some neat little tricks like this, esp. with tan and the inverse functions... and it saves you having to memorize hundreds of different basic integrals and derivatives.

Answer 5

the answer would be just Sec(x)+C...
the derivative of Sec(x) is tan(x)sec(x)...

Sometimes you have to memorize these stuff....

Let me give you some others..
tan(x)=sec^2(x)
cos(x)= -sin(x)
sinx= cosx......

Answer 6

It would be by substitution...
tan(x)*sec(x)=sin(x)*cos ^ -2(x)
u = cos (x)
du = -sin(x) dx
-du = sin (x) dx
so...the result is 1/cos(x)!!!!

Answer 7

Sec (x) + c.