Graduate Can a function inside the integral be erased?

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SUMMARY

The discussion centers on the integral equality $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ with the specific function $$f(x)=e^x$$. It concludes that one cannot assert $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$ solely based on the given equality. The reasoning is that while $$\int_a^b f(x)(g(x) - h(x))\,dx = 0$$, it does not imply that $$g(x) - h(x)$$ is zero everywhere on the interval (a, b) without additional conditions on the sign of $$F(x)$$.

PREREQUISITES
  • Understanding of definite integrals and their properties
  • Familiarity with the function $$f(x)=e^x$$
  • Knowledge of the Fundamental Theorem of Calculus
  • Basic concepts of continuity and integrability of functions
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  • Study the implications of the Fundamental Theorem of Calculus on integrals
  • Explore conditions under which integrals equal zero
  • Learn about the properties of continuous functions and their integrals
  • Investigate the concept of almost everywhere equality in integrals
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Mathematicians, calculus students, and educators seeking to deepen their understanding of integral properties and the implications of function equality within integrals.

CECE2
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Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?
 
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CECE2 said:
Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?

So you have \int_a^b f(x)g(x)\,dx - \int_a^b f(x)h(x)\,dx =<br /> \int_a^b f(x)(g(x) - h(x))\,dx = \int_a^b F(x)\,dx = 0. You cannot in general conclude from \int_a^b F(x)\,dx = 0 that F(x) \equiv 0 everywhere on (a,b). You can only reach this conclusion if you know in addition that F(x) \geq 0 everywhere or that F(x) \leq 0 everywhere.
 

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