12 Days of Christmas Integrals: ∫ x^2 ÷ (1 + x^2) dx
On the Fourth day of Christmas Integrals, the integral featured today is...
∫ x^2 ÷ (1 + x^2) dx
Time for a little algebraic manipulation: add and subtract 1 ÷ (1 + x^2)
= ∫ x^2 ÷ (1 + x^2) + 1 ÷ (1 + x^2) - 1 ÷ (1 + x^2) dx
= ∫ (x^2 + 1) ÷ (1 + x^2) - 1 ÷ (1 + x^2) dx
= ∫ (1 + x^2) ÷ (1 + x^2) - 1 ÷ (1 + x^2) dx
= ∫ 1 - 1 ÷ (1 + x^2) dx
= x - arctan(x) + C
Eddie
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