Integration: what variables can you move outside of the integrand?

1. Jan 24, 2013

tahayassen

$$1.\int { x } dx=x\int { 1 } dx\\ 2.\int { t } dx=t\int { 1 } dx\\ 3.\int _{ x }^{ x+1 }{ x } dt=x\int _{ x }^{ x+1 }{ 1 } dt$$

Which of the equations are correct?

2. Jan 24, 2013

Zondrina

3 and 2 are both correct.

3. Jan 24, 2013

Staff: Mentor

And 1 is incorrect. The following is a property of integrals:
$\int k~f(x)~dx = k\int f(x)~dx$, for k a constant, but there is no property that says you can move a variable across the integral sign.

4. Jan 24, 2013

Zondrina

Integral xdx is certainly not the same as x times integral dx.

5. Jan 25, 2013

HallsofIvy

You can move constants (and so variables that are independent of the variable of integration and so are treated like constants in the integration) outside the integral.

No, x is the variable of integration so we cannot take it outside the integral.
The integral on the left is $x^2/2+ C$ and on the right $x(x+ c)= x^2+ cx$.

If we know that t is independent of x, then both integrals are "tx+ C". If t is a function of x then the first is still "tx+ C" but the other depends upon exactly what function of x t is.

If x is independent of the variable of integration, t, both of those are the same and are equal to x(x+1- x)= x. If x is a function of t, then the left depends upon exactly what function of t x is while the right is still x.

6. Jan 26, 2013

tahayassen

Thanks for the clear-up. :)