what is the intigral of sec x dx and also what is the intigral od cos^2x dx?

Answer 1

integral of secx= log!secx+tanx!+C
or log tan!pi/4+x/2!+C
integralof cos^2x=integral of (1+cos2x)/2
=1/2x+1/2sin2x+C

Answer 2

Use the identity of cos^2 x = 1/2 + 1/2 cos 2x

Answer 3

The integral of sec(x) is just something you *have* to know, as the methods for solving it aren't fundamental to the course.

Integral (sec(x))dx = ln |sec(x) + tan(x)|

To solve for the integral of cos^2(x), you have to use the double angle identity.

This identity goes as follows:

cos^2(x) = [1 + cos(2x)]/2

Therefore,

Integral (cos^2(x))dx =

Integral ([1 + cos(2x)]/2)

First, pull out all constants; in this case, due to the 2 on the denominator, it's 1/2.

1/2 * Integral (1 + cos(2x))dx

Now, take the integral normally. Note that it's easy to take the antiderivative of a function if the terms inside are linear; all we have to do is "offset" the result of the chain rule by multiplying by the appropriate constant. Integral of cos(2x) is *like* sin(2x), except for the fact that taking the derivative means multiplying by 2 by the chain rule, so to offset that we multiply by (1/2), making the answer (1/2)sin(2x).

1/2 * [x + [1/2]sin(2x)] + C

Meshing that all together with steps I'm not going to show here,

[2x + sin(2x)]/4 + C

Answer 4

ln|secx+tanx|+c, you can find this using u-substitution with secx+tanx, your du would therefore be (secxtanx+sec^2 x)dx=integral of du/u which would be ln|u|+c then replace u with secx+tanx.then for cos^2 xdx you would integrate by parts. Or you could find it on a table.

Answer 5

Use integration by parts for sec x dx
log(cos(x/2) + sin(x/2)) - log(cos(x/2)-sin(x/2))

(cos x)^2 dx
1/2(x + cos(x)sin(x))