Can someone check/help with these anti-derivatives? I understand the basics of anti-derivatives, and have been given the following questions:
1: $ \int [t^3 - 5e^t - 4\cos(t)] dt $
My answer:

$ \frac{t^4}{4} - e^{5t} + \sin(4t) + c $

2: $ \int [\frac{8}{x} + 5\sec(x)\tan(x)] dx$
My answer:

$8\ln(x) + \sec^2(5x) + c$

3: $ \int \frac{4x}{(x^2 +5)^3} dx $
My answer: 

None yet, slightly confused since it's a quotient. 

4: $ \int [x^2 \sqrt{2x^2 + 5}]  dx $
My answer:

None yet, confused as well.

So I'm wondering if someone can check number 1 & 2, and then explain/show 3 & 4?
EDIT
These are what I get when I redo them (I've done my best to just look at suggestions rather than answers):
1: $ \frac{t^4}{4} - 5e^t - 4sin(t) + c $
2: $ 8ln(x) + 5sec(x) + c $
3: $ 2 \frac{(x^2 + 5)^{-2}}{-2}$
4: $ \frac{1}{6} \frac{(2x^3 + 5)^{3/2}}{3/2}$
 A: $1$ and $2$ are not quite right. You seem to understand the idea but are missing one thing: 
The derivative of for example $\sin(4t)$ by the chain rule would be $4 \cos(4t)$ not $4 \cos(t)$.  This should show you why you shouldn't bring the constant inside the function you're taking the antiderivative of.
In general if you have a constant multiplied by a function you should take the antiderivative as if the constant were not there, and then multiply that by the constant.
For example for $4 \cos(t)$, we'll ignore the $4$ and take the antiderivative of $\cos(t)$ which is $\sin(t)$, then multiply it by $4$ to get $$ 4 \sin(t).$$
To check that this will work you should take the derivative and make sure its the what you want it to be! 
You have made this error a couple if times in the first two and I advice you to try and find all the places you have and fix it. 
For problem $3$ I advise you to try the $u$-sub $ u = x^2 + 5$.
And for problem $4$ I think the integral is intended to be $x^2\sqrt{2x^3 + 5}$ in which case I would suggest the $u$-sub $u = 2x^3 + 5$. 
A: First of all, you can always check whether an antiderivative is right by differentiating and checking whether you get what you started with.
For 1: Remember that $\int a f(x) \ dx = a \int f(x) \ dx$, that $\int e^x \ dx = e^x + C$ and that $(\sin x)' = \cos x$, not $-\cos x$. Also look at the derivative of $\sin(4x)$ and compare it with the derivative of $4 \sin x$.
For 2: What's the derivative of $\sec^2 x$, and what's the derivative of $\sec x$?
For 3: Try a substitution $u = x^2 + 5$.
For 4: Are you sure that's what you're supposed to do? That integral is pretty difficult, certainly not what I'd expect someone who's just starting with this to be able to do.
