Integrating a Definite Integral with a Dummy Variable

[zorro]

If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?

 

[HallsofIvy]

What do you mean by "substituting t as x"? Certainly after the integration, you have
$$\int_0^x sin(t)dt= cos(t)\right]_0^x= -cos(x)$$

 

[Stephen Tashi]

Do you mean set t = x and take the anti-derivative and forget about the limits of integration? No. It might work by accident in some problems, but in general, you can't do that.

Consider $$\int_0^x e^t dt$$