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Tyrese Pruitt

Tyrese Pruitt

Answered question

2022-06-02

If 0 π 4 tan 6 ( x ) sec ( x ) d x = I then express 0 π 4 tan 8 ( x ) sec ( x ) d x in terms of I

Answer & Explanation

Sanaa Hoover

Sanaa Hoover

Beginner2022-06-03Added 1 answers

Short answer: intergration by parts.
Long answer: denote the integral you want to find as J. Then you have
J = 0 π 4 tan 8 ( x ) sec ( x ) d x = 0 π 4 tan 7 ( x ) d ( sec ( x ) ) = tan 7 ( x ) sec ( x ) | 0 π 4 0 π 4 sec ( x ) d ( tan 7 ( x ) ) = 2 7 0 π 4 tan 6 ( x ) sec 3 ( x ) d x = 2 7 0 π 4 tan 6 ( x ) sec ( x ) ( 1 + tan 2 ( x ) ) d x = 2 7 I 7 J = J
2 7 I 7 J = J
Hope all the steps are clear.

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