How to find Find F' of #int_x^10(t+1/t)dt# Using the Fundamental Theorem of Calulus?

Question

Find F' of #int_x^10(t+1/t)dt# then check the result by first integrating and then differentiating. please help.

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Answer 1 · 2018-04-03T23:52:07

#F'=-x-1/x#

The fundamental theorem of calculus states that if #F(x)=int_a^xf(t)dt# then #F'(x)=f(x)#.

So

#F=int_x^10t+1/tdt=-int_10^xt+1/tdt=-x-1/x#

We check this by integrating first and then differentiating

#F(x)=int_x^10t+1/tdt=[t^2/2+ln|t|]_x^10=50+ln10-(x^2/2-ln|x|)=-x^2/2-ln|x|+50+ln10#

So #F'(x)=d/dx(-x^2/2-ln|x|+50+ln10)=-x-1/x#