Evaluate the integral. ∫012ln⁡(x)xdx

Jean Blumer

Jean Blumer

Answered

2021-12-19

Evaluate the integral.
012ln(x)xdx

Answer & Explanation

turtletalk75

turtletalk75

Expert

2021-12-20Added 29 answers

Step 1
Consider the provided integral,
012ln(x)xdx
Evaluate the above integral.
Step 2
Now, solve the above integral using by part formula.
012ln(x)xdx=201ln(x)xdx
=2[2xln(x)2xdx]01
=2[2xln(x)4x]01
=2(-4)
=-8
sirpsta3u

sirpsta3u

Expert

2021-12-21Added 42 answers

2ln(x)xdx
Lets
nick1337

nick1337

Expert

2021-12-28Added 573 answers

The integration process can be simplified by making a change of variables:
t=x,x=t2,dx=2tdt
Then the original integral can be written as follows: The
4ln(t2)dt
4ln(x2)dx
formula for integration by parts:
U(x)dV(x)=U(x)V(x)V(x)dU(x)
Put
U=4ln(x2)
dV=dx
Then:
dU=8xdx
V=x
Therefore:
4ln(x2)dx=4xln(x2)8dx
Find the integral
8dx=8x
Answer:
4ln(x2)=4xln(x2)8x+C
or
4ln(x2)=4x(ln(x2)2)+C
To write the final answer, it remains to substitute sqrt (x) instead of t.
4x(ln(x)2)+C
Let's calculate a definite integral:
012ln(x)xdx=(4x(ln(x)2))|01
F(1)=-8
F(0)=
I=-8-(0)=-8

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