How to integrate ∫ d x x x 4 − 1

rigliztetbf

rigliztetbf

Answered

2022-06-24

How to integrate d x x x 4 1

Answer & Explanation

Ryan Newman

Ryan Newman

Expert

2022-06-25Added 26 answers

So if you write
tan 1 x 4 1 = θ ,
Then
tan θ = x 4 1 .
A right triangle that tells this story has θ as one angle, x 4 1 as the opposite side and 1 as the adjacent side. Using Pythagorean theorem we can work out the length c of the hypotenuse:
( x 4 1 ) 2 + 1 2 = c 2
which shows that c = x 2
So sec θ = x 2 / 1, that is sec 1 ( x 2 ) = θ = tan 1 ( x 4 1 ) .
Zion Wheeler

Zion Wheeler

Expert

2022-06-26Added 11 answers

We have,
d x x x 4 1
Let x 2 = sec ( θ ) 2 x d x = sec ( θ ) tan ( θ ) d θ
Which further implies that, d x = sec ( θ ) tan ( θ ) 2 d θ
After substitution, the given integral changes to:
1 2 sec ( θ ) tan ( θ ) sec ( θ ) sec 2 ( θ ) 1 d θ
= 1 2 sec ( θ ) tan ( θ ) sec ( θ ) tan 2 ( θ ) d θ
= 1 2 sec ( θ ) tan ( θ ) sec ( θ ) tan ( θ ) d θ
= 1 2 d θ
= 1 2 θ + C
= 1 2 sec 1 ( x 2 ) + C

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