Prove.prove that integral from a to b x dx=(b^2-a^2)/(2) using the definiton of a Definite Integral.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that using the following formula you may evaluate the given integral such that:

`int x dx = x^2/2 + c`

You need to remember the fundamental theorem of calculus that helps you to evaluate definite integrals such that:

`int_a^b f(x) dx = F(b) - F(a)`

Reasoning by analogy yields:

`int_1^b x dx = x^2/2|_a^b`

`int_1^b x dx = b^2/2 - a^2/2`

Factoring out `1/2`  yields:

`int_1^b x dx = (1/2)(b^2 - a^2) => int_1^b x dx =(b^2 - a^2)/2`

Hence, evaluating the given definite integral, using the fundamental theorem of calculus, yields `int_1^b x dx =(b^2 - a^2)/2.`

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