Integration help!!!!!?

Question

I need help integrating:

1/cos^2(y) with respect to y.

1/(2x^(1/2)) with respect to x.

I need to know the technique to solve these. I am kind of rusty on these things.

thanks!

Answer 1

Integral (1/[cos^2(y)]) dy

To solve this one, use trig identities. Note that 1/cos(y) = sec(y), so we're going to have sec^2(y).

Integral (sec^2(y)) dy

This is a known derivative; it's the derivative of tan. Therefore, our answer is

tan(y) + C

2) Integral ( 1 / [2x^(1/2)] ) dx

One thing you should do is pull out all constants prior to integrating. This is a really useful (and valid) step.

(1/2) * Integral (1/[x^(1/2)]) dx

Note that 1/(x^a) is the same as x^(-a), so we have

(1/2) * Integral (x^(-1/2)) dx

Now, we just use the reverse power rule. Recall that the antiderivative of x^a (for a not equal to -1) is [x^(a + 1)]/(a + 1), so

(1/2) [x^(-1/2 + 1)] / (-1/2 + 1)

(1/2) [x^(1/2)] / (1/2)

Simplifying that complex fraction, we have

(1/2) (2) (x^(1/2)) + C

or just simply

x^(1/2) + C

Answer 2

The integal of x^a if is not -1 is 1/(a+1) * x^(a+1)

Here you have 1/2* x^(-1/2) so the integral is = 1/2 *( 1/(-1/2+1) *x^1/2= x^1/2
1/cos^2(y). the derivative of tg y = sen/cos is = 1/cos^2 *(cos^2+sin^2) =1/cos^2 so the integral is tg(y)

Answer 3

The first one is known it's just tan y. If you don't know some derivatives you'll have trouble integrating...

For the secund one it's x^{1/2}. which you would see immediately if you were used to derivatives if powers of x.... Don't worrry it will come

Answer 4

1/cos^2(y)=sec^2(y)
after integration=tan(y)
1/(2x^(1/2))=(x^-1/2)/2
after integration = -(x^1/2)/4
we know that x^n
after integration= x^n+1/n+1