Integrate logarithmic function int_0^(oo) (ln(1+25x^2))/(1+16x^2)dx

c0nman56

c0nman56

Answered question

2022-10-15

Integrate logarithmic function 0 ln ( 1 + 25 x 2 ) 1 + 16 x 2 d x

Answer & Explanation

exalantaswo

exalantaswo

Beginner2022-10-16Added 14 answers

Let tan t = 4 x. Then,
0 ln ( 1 + 25 x 2 ) 1 + 16 x 2 d x = 1 4 0 π / 2 ln ( 1 + 25 16 tan 2 t ) d t = 1 4 I ( 25 16 )
where
I ( a ) = 0 π / 2 ln ( 1 + a tan 2 t ) d t , I ( a ) = 0 π / 2 tan 2 t d t 1 + a tan 2 t = π 2 a ( a + 1 )
Thus,
0 ln 1 + 25 x 2 1 + 16 x 2 d x = 1 4 I ( 25 16 ) = 1 4 0 25 16 I ( a ) d a = π 8 0 25 / 16 d a a ( a + 1 ) = π 4 ln ( a + 1 ) | 0 25 16 = π 4 ln 9 4

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