# Thread: integrals yielding inverse trigonometric functions

1. ## integrals yielding inverse trigonometric functions

according to back of book

Thanks alot!

2. Basically use the substitution t= ln x
And the identity for inverse tangent functions.
Shown below.

I don't agree with the book either . . .

The answer should be: according to back of book

. . . . . . . . . . . .1 . . . . dx
We have: .∫ -------------·---
. . . . . . . . .1 + (ln x)² . x

. . . . . . . . . - . . . . . . dx
Let u = ln x . . du = ---
. . . . . . . . . . . - . - . . .x

. . . . . . . . . . . du
Substitute: .∫ -------- . = . arctan(u)
. . . . . . . . . . 1 + u²

. . . . - . . . . - . . . - . . . . . . .|e
Back-substitute: .arctan(ln x) |
. . . . - . . . . - . . . - . . . . . . .|1

Evaluate: . arctan(ln e) - arctan(ln 1)

. . . . . .= . arctan(1) - arctan(0)

. . . . . .= . ¼π - 0 . = . ¼π

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It appears that the author didn't back-substitute.

I substituted for the derivative rather than original function.

It seems the author made the same mistake I did.

5. Originally Posted by Soroban

I don't agree with the book either . . .

. . . . . . . . . . . .1 . . . . dx
We have: .∫ -------------·---
. . . . . . . . .1 + (ln x)² . x

. . . . . . . . . - . . . . . . dx
Let u = ln x . . du = ---
. . . . . . . . . . . - . - . . .x

. . . . . . . . . . . du
Substitute: .∫ -------- . = . arctan(u)
. . . . . . . . . . 1 + u²

. . . . - . . . . - . . . - . . . . . . .|e
Back-substitute: .arctan(ln x) |
. . . . - . . . . - . . . - . . . . . . .|1

Evaluate: . arctan(ln e) - arctan(ln 1)

. . . . . .= . arctan(1) - arctan(0)

. . . . . .= . ¼π - 0 . = . ¼π

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It appears that the author didn't back-substitute.
OK lets try integrating this numericaly. I won' go into the detail of
how to do it, but the answer is ~=0.785, while the quoted answer is
numericaly ~=0.433.

RonL

6. ## Really sorry

Made a Mistake again been careless with posting lately...
sorry and ill try to post correctly next time..................

7. Just in case

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