Integation (1 Viewer)

[joesmith1975]

need help

the intergal of cos x /1-sin^2x dx

Thanks

 

[Riviet]

Let u = sin x
du = cos x dx

Therefore, your integral becomes du/(1-u²). Then you can factorise the denominator and split the fraction into two using partial fractions.

 

[joesmith1975]

the subtitution method is not very good and i didnt get the right answer. i was taught a different way that no one is taught which is called the "sulpy method"

 

[joesmith1975]

the answer is 1/2 log (1+sinx)/(1-sinx)

 

[Riviet]

I knew there would have been an easier way rather than a subsitution, Iruka's method is the way to go.

 

[joesmith1975]

not the right answer thou

 

[joesmith1975]

yere u prove i minus the other equals 0 but if i didnt give the answer how u show LHS=RHS

 

[joesmith1975]

then by use of ur method by subing in the trig. values then :-

the intergal of tan^3x.sec^2x dx would be

intergal of tan^3 x.tan^2 x+1 dx

Intergal of tan^5 x +tan^3 x dx

=tan^6 x/6 + tan^4 x/4 +C

which its the answer only tan^4 x/4 is the answer

method doesnt work.

 

[icycloud]

LHS = ln(secx+tanx) = ln(1/cosx + sinx/cosx)
= ln((1+sinx)/(cosx))
= ln((1+sinx)/(sqrt(1-sin²x))
= ln((1+sinx)/(sqrt((1-sinx)(1+sinx)))
= ln(sqrt((1+sinx)/(1-sinx))
= 1/2 ln((1+sinx)/(1-sinx))
= RHS

 

[icycloud]

then by use of ur method by subing in the trig. values then :-
the intergal of tan^3x.sec^2x dx would be
intergal of tan^3 x.tan^2 x+1 dx
Intergal of tan^5 x +tan^3 x dx
=tan^6 x/6 + tan^4 x/4 +C
which its the answer only tan^4 x/4 is the answer
method doesnt work.

Let u = tanx, then du=sec²x dx
so the integral becomes u^3 du = u^4/4 + C = tan⁴x/4 + C

 

[joesmith1975]

thanks

 

[icycloud]

BTW, using partial fractions:

u = sinx, du=cosx dx

Integral = du/(1-u^2) = du/((1+u)(1-u))

1/((1+u)(1-u)) = A/(1+u) + B/(1-u)
1 = A(1-u) + B(1+u)
When u = 1, B = 1/2
u = -1, A = 1/2

Therefore, integral becomes 1/2 [du/(1+u) + du/(1-u)]
= 1/2[ln|1+u| - ln|1-u|] + C
= 1/2 ln | (1+sinx)/(1-sinx) | + C

(This is how you get your textbook's answer.)