Finding a function (f) and a number a that satisfy the definite integral

**AnElephant09** · January 23rd 2013, 03:11 PM

Question: Find a function (f) and a number 'a' that satisfy the definite integral
Here is the integral: 6+∫[a,x,f(t)/t^2,t]=2sqrt(x) which comes out to

Can somebody please explain how to do this? i tried using the FToC parts 1 and 2, but i think im using them wrong here...
Thanks in advance!

**johng** · January 23rd 2013, 04:24 PM

Assume f is continuous. Then the derivative of $\int_a^x {f(t)\over t^2}dt$ is ${f(x)\over x^2}$ , which must then be ${1\over \sqrt{x}}$ -- I took the derivative of both sides of the equation. So $f(x)={x^2\over \sqrt{x}}=x^{3/2}$ . Now $\int_a^x {t^{3/2}\over t^2}dt=\int_a^x t^{-1/2}dt=2t^{1/2}|_a^x=2\sqrt{x}-2\sqrt{a}$ . Finally then $6+2\sqrt{x}-2\sqrt{a}=2\sqrt{x}$ implies $3=\sqrt{a}$ or $a=9$ .

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Finding a function (f) and a number a that satisfy the definite integral

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