The functions $f$ and $g$ are inverse and monotonically increasing. Determine what the following expression is equal to! $$ \int_a^b f(x)\, dx + \int_{f(a)}^{f(b)} g(x)\, dx $$
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2I would start by sketching a graph and then colouring in what the integrals represent in that picture. – Matti P. Jan 17 '24 at 08:58
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Asked and answered here: https://math.stackexchange.com/q/1115222/42969 – Martin R Jan 17 '24 at 11:17
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More identical or similar questions: https://math.stackexchange.com/questions/linked/1115222?lq=1 – Martin R Jan 17 '24 at 11:18
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This is integration by parts. See for example the picture here.
Formally you substitute $f(y)=x$ and get $$\int_{f(a)}^{f(b)} g(x)\, dx=\int_a^bg(f(y))f'(y)dy=\int_a^b yf'(y)dy$$ hence $$\int_a^b f(x)\, dx + \int_{f(a)}^{f(b)} g(x)\, dx=\int_a^b(x') f(x)\, dx + \int_a^b xf'(x)dx=f(b)b-f(a)a$$ where in the last step we used the integration by parts formula (or the product formula for the derivative).