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

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

1 Answer
Apr 3, 2018

F'=-x-1/x

Explanation:

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