Show that the derivative of f(x) =xln(x)−x is ln(x). Use this to compute int ln(x)dx.

Grayson Pierce

Grayson Pierce

Answered question

2022-07-16

Show that the derivative of
f ( x ) = x ln ( x ) x
is ln ( x ).
Use this to compute ln ( x ) d x .

Answer & Explanation

Damarion Pierce

Damarion Pierce

Beginner2022-07-17Added 11 answers

Sum/difference rule of differentiation,
( f ± g ) = f ± g
The derivative of x ln x x
d ( x ln x x ) d x = d ( x ln x ) d x = d ( x ) d x = d ( x ) d x ln x + d ( ln x ) d x x d ( x ) d x
(  derivative of x w.r.t.x is 1, ferivative of  ln x  w.r.t.x is  1 x ) d ( x ln x x ) d x = ln x + 1 1 = ln x
Now, ln x d x
Using integration by parts,
u = ln x , d v = d x
Then, d u = 1 x d x , x = v
This gives, ln x d x = u d v = u v v d u = ln x x x 1 x d x = x ln x x + c, where c is a constant.

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