Trying to get my head around integration?

Question

∫(4x^3 + 7x^-2) dx

= 4x^4/4 + 7x^-1/-1 + C

I understand the above. I always get that far on my workings.

'= x^4 - 7/x + C' However, I do not understand this part.

I imagine it's very simple and right in front of me and I'm looking too hard at all my content. 

Been studying differentiation for day and just started looking integration for the first time in memory. A lot to take in in short time. Tia

Update:

Actually not a troll. Just about 14 straight hours on this and feeling a little brain fried. 

Update 2:

I wish I could delete the question now as I feel stupid. Thanks everyone.

Answer 1

Favourite answer

If it helps, this is how I think of getting integrals of polynomial-like expressions.

Remember that this is the reverse of a derivative.  The steps for that is, for each term:

ax^b  <-- "old" pre-derivative term

The new coefficient is the product of the old coefficient and old exponent.

The new exponent is the old exponent minus 1.

Any constant terms are dropped.

So the above term becomes:

abx^(b - 1)

When getting the integral (or anti-derivative) you reverse those steps.

Starting from the same place: ax^b

Start with adding 1 to the old exponent to make the new exponent.

Then divide the old coefficient by the new exponent to get the new coefficient.

A new constant term is added.

[a / (b + 1)] x^(b + 1) + C

So in your example:

∫(4x³ + 7x⁻²) dx

Each term has you add 1 to the exponent (3 becomes 4 and -2 becomes -1), then divide the 4 by the 4 (equal to 1) and the 7 by the -1 (to be -7), and add the constant term:

x⁴ - 7x⁻¹ + C

What I have above is the simplified form of what you had as part of your question.  Hope this helped.

Answer 2

The 4x^4/4 means (4/4)x^4 = 1 x^4 = x^4.

The 7x^(-1)/-1 means (7/-1) x^(-1) = -7 x^(-1) = -7 (1/x) = -7/x.

If you have not mastered elementary algebra, you have no business being in a calculus class.

Answer 3

If you can differentiate the (new) equation to get 4x^3 + 7x^-2, then you have it right.

x⁴ differentiates to 4x³

7x⁻¹ differentiates to -7x⁻²

so x⁴ -7x⁻¹ +C

Answer 4

Hmmm, sounds like a troll, on many levels.  But in case you are not:

4x^4/4 = 4 * (x^4)  / 4 = (4 / 4) * x^4 = 1 * x^4 = x^4. 

IOW, a*b*c = (a*c) * b. Remember?

Same idea for your other term. 

Done!

OK, not a troll. But since your question related to elementary school math (or at best early highschool) , and that you are anon, your question does not look sincere, hence the "troll-like" question. Now you know more of some characteristics of trolls.