# Applying integration to math problems

Gold Member
Homework Statement:
I am looking at the integration of ##(x+2)^2## with respect to ##x##
Relevant Equations:
Integration
Ok i know that,
##\int (x+2)^2 dx= \int [x^2+4x+4] dx= \dfrac{x^3}{3}+2x^2+4x+c##

when i use substitution;

i.e letting ##u=x+2## i end up with;

##\int u^2 du= \dfrac{u^3}{3}+c=\dfrac {(x+2)^3}{3}+c=\dfrac{x^3+6x^2+12x+8}{3} +c##

clearly the two solutions are not the same...

appreciate your insight...which approach is more concrete? note that when we differentiate both solutions we get the same function i.e ##x^2+4x+4##.

• PeroK and malawi_glenn

Gold Member
clearly the two solutions are not the same...
They are the same to within a constant which is what one expects. Try integrating with bounds from 0 to a.

• • scottdave, SammyS, Mark44 and 1 other person
Alex Schaller
Homework Statement:: I am looking at the integration of ##(x+2)^2## with respect to ##x##
Relevant Equations:: Integration

Ok i know that,
##\int (x+2)^2 dx= \int [x^2+4x+4] dx= \dfrac{x^3}{3}+2x^2+4x+c##

when i use substitution;

i.e letting ##u=x+2## i end up with;

##\int u^2 du= \dfrac{u^3}{3}+c=\dfrac {(x+2)^3}{3}+c=\dfrac{x^3+6x^2+12x+8}{3} +c##

clearly the two solutions are not the same...

appreciate your insight...which approach is more concrete? note that when we differentiate both solutions we get the same function i.e ##x^2+4x+4##.
Indefinite integrals can be regarded as a set (family) of curves, and each of the curves can be obtained by shifting in a parallel the curve, upwards or downwards (along the "Y" axis).

• chwala
Homework Helper
Gold Member
The constant "c" in the first example is not necessarily the same "number" as the constant in the second example.

• chwala
Homework Helper
Gold Member
2022 Award
clearly the two solutions are not the same...
Why are they not the same? 8/3 + c is a constant right?

You wrote that upon differentiating, we get the same original function.
Well, that is the definition of indefinite integral, i.e. primitive function my friend.

• chwala
Gold Member
Why are they not the same? 8/3 + c is a constant right?

You wrote that upon differentiating, we get the same original function.
Well, that is the definition of indefinite integral, i.e. primitive function my friend.
True, that ought to have been pretty obvious to me... ...cheers man!