Integrals of these functions

[TwinkleTits]

I've a maths test coming up and integration is the area im having trouble with. I've been practicing but I still dont really know what the hell Im doing Im just following methods. Heres how id approach these questions they dont give the answers though so I dunno if Im right or not. Can someone tell me how Im doing.

1) (x + 3)^2 dx
All I'd do here is multiply it out and get a polynomial. Dead simple Im assuming.

2.) 7/x dx
I'd just convert this into 7lnx

3.) cos8x dx
This one I dunno about. In differentiation I'd convert it to (8)(-sin8x). In integration would I divide by that 8 instead of multiply? eg. sin8x/8?

4.)(sqroot(t)) + 1/t^2 dx
First I'd convert sqroot(t) into t^1/2. Then convert the 1/t^2 to t^-2. Then I'd have something I could integrate. Is that right?

Now they get hard.
5.) (DEFINITE WITH LIMITS 2 and 1) (x^2) / (x^3 + 3) dx
The only method I've learned so far is substitution so I'd try it on this. I let u = x^3 + 3. I know that dy/dx is 3x^2 so du = (3x^2)(dx). Ah thats as far as I can get Im lost with this one.

6.) (DEFINITE WITH LIMITS PI/4 and 0) cos^2 x dx
Holy fuck.

[lolocaust]

1 through 4 are right
5 is ln(x^3 + 3) / 3 (chain rule puts the x^2 on top after deriving)
then you just plug in the numbers and do whatever

I forget six, I'll reply again if I figure it out

[TwinkleTits]

Nice one lolocaust. You sorted my out for a few differentiation questions too.

[lolocaust]

for 6 you need to know that (cos(x))^2 = (1/2)(1+cos(2x))

and then just integrate that mess

[harry_hardcore_hoedown]

Originally Posted by TwinkleTits

1) (x + 3)^2 dx
All I'd do here is multiply it out and get a polynomial. Dead simple Im assuming

When it's factorized in the form a(bx-q)^n it's simpler than that. The antiderivative is [a/b(n+1)](bx-q)^(n+1) + c. You can do it your way, and it's preferable when you're doing a definite integral and need to evaluate, but if not, and you're lazy you don't need to bother expanding it.

[Mantikore]

yep, 5 requires you to see that the numerator is a bit like the derivative of the denominator. multiply the numerator by 3 to get the exact derivative of it, then divide it by three on the outside of the integral. then just use the natural log thing.

for question 6, yeah you have to know an identity for it. im not sure about what your school/university is like, some people get a table of standard integrals to refer to.

however, we can derive this
remember how cos(2x) can be written like a difference of two squares?
cos(2x) = (cos(x))^2-(sin(x))^2)................(i)

and you know that
(cos(x))^2 + (sin(x))^2) = 1
(sin(x))^2 = (cos(x))^2) + 1..........substitute that into equation (i), then rearrange to isolate (cos(x))^2
cos(2x) = (cos(x))^2 - (1- (cos(x))^2))
cos(2x) = (cos(x))^2 - 1 + (cos(x))^2
cos(2x) = 2(cos(x))^2 - 1
(cos(x))^2 = (cos(2x) + 1) /2
(cos(x))^2 = cos(2x)/2 + 1/2..................then integrate dat' mofo