How to solve certain kind of integral

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The discussion focuses on solving double integrals of the form $$\int_a^b \int_0^{f(x)} g(r) dr dx$$. User ariberth provides a solution that involves applying the Fundamental Theorem of Calculus, resulting in the expression $$\int_a^b G(f(x)) \, dx - G(c) \cdot (a-b)$$. This highlights the importance of correctly applying limits to constants in integral calculus. The conversation emphasizes the need for careful manipulation of integral limits to achieve accurate results.

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

ariberth
 
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Ah ok i think i found it already :)

$$\int_a^b \int_c^{f(x)} g(r)dr dx = \int_a^b |_c^{f(x)} G(r) = \int_a^b G(f(x)) - G(c) dx = \int_a^b G(f(x))dx - G(c) $$
 
You forgot to apply the limits to the constant $G(c)$:

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

:)
 
Haha yes, thank you :)
 

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