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uplakanimkk 2022-07-05 Answered
Solving x 1 x 2 d x E U 0 tan ( a x ) 2
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Answers (1)

Marisol Morton
Answered 2022-07-06 Author has 13 answers
Well, we are trying to find the following integral:
(1) I n ( α , β ) = 1 α + β tan 2 ( n x )   d x
Let's substitute u = n x, this leads to:
(2) I n ( α , β ) = 1 n 1 α + β tan 2 ( u )   du
Let's substitute s = sin ( u ) , this leads to:
(3) I n ( α , β ) = 1 n 1 α + s 2 ( β α )   ds
Let's substitute w = s β α 1 , this leads to:
(4) I n ( α , β ) = 1 n β α 1 w 2 + 1   dw
This is a very standard integral, which gives:
(5) I n ( α , β ) = 1 n β α ln | w + w 2 + 1 | + C
So, we end up with:
(6) I n ( α , β ) = 1 n β α ln | sin ( n x ) β α 1 + sin 2 ( n x ) | β α 1 | + 1 | + C

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