How to find the anti-derivative of this?

Question

moongazer · Answer

$$\int\limits (5t+2)^4 dt$$

my teacher said you can't directly use the formula:

[u^(n+1)]/[n+1] + c

but in the problem

$$\int\limits (x-2)^2 dx$$

she  just let me use the formula

isn't it that its just the same?

anonymous · Answer

No. The derivative of (x-2) is just 1. The derivative of  (5t+2) is 5. so you have to account for that 5. So you would need to pull out 1/5 in front of the integral to account for that 5.

So when integrating (x-2)^4 dx, then you can use the rule  of u^n...as u = x -2, du = dx.

But in the integral of (5t+2), if u = 5t + 2, du = 5 dt...and dt = 1/5 du, and that is where you are "pulling out" the 1/5.

anonymous · Answer

So...in essence, if u = any function of x, and du = dx, then you can use the rule for integrating u^n as u^(n+1)/(n+1).

moongazer · Answer

I think I already got the idea but still didn't fully understand the concept.

how about if it is something like $$\int\limits (x^3+2)^4 dx$$

anonymous · Answer

If u = (x^3 + 2), then du = 3x^2 dx....so you are stuck.

anonymous · Answer

so you cannot use the rule of integral u^n.

moongazer · Answer

so you can use the u^n integral rule 
only for the integrand in the form something like (kx +c)^n ?