How to find Find F' of $\int_{x}^{10} (t + \frac{1}{t}) d t$ $int_x^10(t+1/t)dt$ Using the Fundamental Theorem of Calulus?

Question

Find F' of $\int_{x}^{10} (t + \frac{1}{t}) d t$ $int_x^10(t+1/t)dt$ 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 $F'(x)=d/dx(-x^2/2-ln|x|+50+ln10)=-x-1/x$