Evaluate the following indefinite integral

Question

Evaluate the following indefinite integral:

 

∫x*secx²*x dx

Answer 1

Since this is a product of 2 functions you should try integration by parts....∫u*dv = uv-∫v*du

 

So let u=x....that make du = dx.......and make dv = sec²x......so that v = tan x.

 

So ∫x*sec² x dx = x*tan x - ∫tan x dx

 

So now the integral of tan x can be done with a simple u substitution by considering ∫ sin x / cos x dx.

let u = cos x.....making du = -sin x dx. your integral becomes ∫ - 1/u du which equals - ln ⌈u⌈ = - ln ⌈cos x⌈...this is absolute value but I don't have a symbol for that. So substituting that back in

 

∫ x*sec² x dx = x*tan x + ln ⌈cos x⌈ + c......you can take the derivative and see if you get the original if you'd like.

 

The derivative would be (1)tan x + x sec² x + 1/cos x * -sin x

                      which is    tan x   + xsec2 x - tan x

                      which is just x*sec² x.........so the answer is correct.

 

Hope this helped.