Question

How do you find the antiderivative of `3x^2 + sin(4x)+tan x sec x`?

Answer 1

The idea is to use a substitution. The notation varies, depending on what notation you've learned up to this point.

Be cause you are saying "antiderivative", rather than "integral" , I will assume that you haven't learned integral notation yet.

Part 1 Not too tough
So, the antiderivative of `x` to a power is the opposite of the derivative of `x` to a power. For the derivative, you multiply by the exponent, then make a new exponent by subtracting `1`. The opposite, then if first make a new exponent by adding `1` and then dividing (opposite of multiplying) by the new exponent.

So the antiderivative of `x^2` is `(x^3)/3`.

When we find derivatives any constant we're multiplying by just stays out front. It's the same for the antiderivative. So the antiderivative of `3x^2` is `3(x^3)/3`. Which simplifies to just `x^3`. (Check by differentiating `x^3`, Yes, we do get `3x^2`.

Part 2 Also not too tough
The antiderivative of `tanxsecx` is a function whose derivative is equal to `tanxsecx`. Think through the derivatives of trig functions you've memorized, `d/(dx)(sinx)=cosx` and so on. When you get to `d/(dx)(secx)=secxtanx` Stop!

`secxtanx=tanxsecx` so the antiderivative of `tanxsecx` is`secx`

Part 3 A little tougher, but you'll get used to it
(Really not bad, when you learn more notation.)
Now, for the middle, we need a function whose derivative is `sin(4x)`
There are a few ways of describing this next bit, here's one of them:

I've just thought about derivatives of trig function and one of them is `d/(dx)cosx=-sinx`. That's not exactly what I need, but it's close.

Why is it not right? Two reasons. We don't want the minus sign and we want `4x`, not just plain `x`.

So, what's the derivative of `cos(4x)`? It's`( -sin(4x))*4`

That's pretty close. In fact if we get rid of the minus sign and the extra `4`, we'll have it. Remember about the constant just staying out front when we differentiate?
Well, that mean is I use `1/4cos(4x)` the `4`'s will cancel after i find the derivative. To get rid of the minus, we'll stick a minus out front as well.

`d/(dx)(-1/4cos(4x))=sin4x`

Put the pieces together and add + C

The antiderivative of `3x^2+sin(4x)+tanxsecx` is `x^3-1/4cos(4x)+secx+C`

Wehn you learn other notation, you'll learn to call this "integration by substitution" -- though it will look a little different.