Show that the value of a definite integral is unity \int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx=1

adiadas8o7

adiadas8o7

Answered question

2022-04-30

Show that the value of a definite integral is unity
24log(9x)log(9x)+log(3+x)dx=1

Answer & Explanation

Volsa280

Volsa280

Beginner2022-05-01Added 16 answers

Proof. By making a substitution y=6x, we get
I=42f(6y)f(y)+f(6y),dy=24f(6y)f(y)+f(6y),dy
Therefore
2I=24f(x)f(x)+f(6x)dx+24f(6y)f(y)+f(6y)dy
=24f(x)f(x)+f(6x)dx+24f(6x)f(x)+f(6x)dx=241,dx=2

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