## Sunday, December 26, 2021

### 12 Days of Christmas Integrals: ∫ (x ∙ arctan(x)) ÷ (1 + x^4) dx

12 Days of Christmas Integrals:  ∫ (x ∙ arctan(x)) ÷ (1 + x^4) dx

On the Second day of Christmas Integrals, the integral featured today is...

∫ (x ∙ arctan(x)) ÷ (1 + x^4) dx

This integral can be approached by using the substitution method:

Let u = arctan(x^2).  Then:

du = 1 ÷ (1 + x^4) ∙  d/dx(x^2)

du = 1 ÷ (1 + x^4) ∙  (2  ∙ x) dx

1/2 du = x ÷ (1 + x^4) dx

Then:

∫ (x ∙ arctan(x)) ÷ (1 + x^4) dx

= ∫ u ∙ 1/2 du

= 1/2 ∙ ∫ u du

= u^4/4 + C

Substitute back:

= (arctan(x^2))^2 / 4 + C

Eddie

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