Question

Integral:secxdx=ln(secx+tanx)+C
(a) Write sec x as cos x/(1 − sin2 x) and make a substitution to simplify the integral. (b) Integrate the resulting rational function by the partial fractions method.
(c) Use the identity sin2 x + cos2 x = 1 to show that your result is equal to (1).

Answer 1

try u=1-sin^2x

Answer 2

can you break this problem down for me, i'm very lost

Answer 3

u=1-sin^2x
du=?

Answer 4

cos^2?

Answer 5

dx

Answer 6

ok sorry i got it now

Answer 7

\[\int\frac{\cos(x)dx}{1-\sin^2(x)}\]put \(u=\sin(x), du=\cos(x)dx\) get
\[\int\frac{du}{1-u^2}\] then use partial fractions

Answer 8

\[\frac{1}{1-x^2}=\frac{1}{2(x+1)}-\frac{1}{2(x-1)}\]

Answer 9

integrate term by term, then substitute back
sorry i got confused

Answer 10

so its the last one?

Answer 11

yes, integrate the last thing. then replace \(x\) by \(\sin(x)\)

Answer 12

so where do i go from here? common denominators?

Answer 13

or ln ( 2x+2)=ln(2x-2)