find the integral of

f (x) = (1+x^2)ArcTan[ x]

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- October 17th 2006, 01:44 PMfair_lady0072002please help- integration
find the integral of

f (x) = (1+x^2)ArcTan[ x] - October 17th 2006, 03:10 PMtopsquark
- October 17th 2006, 05:48 PMThePerfectHacker
My, where you do get these complicated integration from?

Here is what you do,

INTEGRAL (1+x^2)^2*atan(x)*(1+x^2)^{-1} dx

Let, u=atan(x) then, u'=(1+x^2)^{-1}

Then,

sec^2 u=1-x^2

cos^2 u=x^2

Then,

sec^2 u +2cos^2 u=1+x^2

Thus,

INTEGRAL (sec^2 u +2cos^2 u)^2 u*u' dx

Substitution rule it is composition of outer function,

INTEGRAL (sec^2 u+2cos^2 u)^2 du

Open parantheses, (secant*cosine=1)

INTEGRAL (sec^4 u+4cos^4 u+2) du

And this is definitely solvable - October 17th 2006, 06:17 PMtopsquark
- October 17th 2006, 07:26 PMThePerfectHacker
- October 17th 2006, 08:43 PMSoroban
Hello, fair_lady0072002!

Quote:

Find the integral of: .(1 + x²) arctan(x) dx

. . Then: .1 + x² .= .1 + tan²(x) .= .sec²(x)

Substitute: . ∫ sec²(u) · u · sec²(u) du . = . ∫ u·sec^4(u) du

. . . Good luck!