How to solve certain kind of integral

[ariberth]

Hello everybody!
Are there methods to solve integrals of the following form?
$$\int_a^b \int _0 ^{f(x)} g(r) dr dx$$

ariberth

 

[ariberth]

Ah ok i think i found it already :)

$$\int_a^b \int_c^{f(x)} g(r)dr dx = \int_a^b |_c^{f(x)} G(r) = \int_a^b G(f(x)) - G(c) dx = \int_a^b G(f(x))dx - G(c) $$

 

[Fantini]

You forgot to apply the limits to the constant $G(c)$:

$$\int_a^b G(f(x)) - G(c) \, dx = \int_a^b G(f(x)) \, dx - G(c) \color{red}{\cdot (a-b)}.$$

:)

 

[ariberth]

Haha yes, thank you :)