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Integration by Substitution

AP Calculus AB· difficulty 3/5

∫x1+x2 dx=\displaystyle\int \dfrac{x}{1 + x^2}\,dx =

  • A

    11+x2+C\dfrac{1}{1 + x^2} + C

  • B

    ln⁡(1+x2)+C\ln(1 + x^2) + C

  • C

    12ln⁡(1+x2)+C\dfrac{1}{2}\ln(1 + x^2) + C

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  • D

    arctan⁡x+C\arctan x + C

Explanation

u=1+x2u = 1 + x^2, du=2x dxdu = 2x\,dx. ∫(1/u)du/2=ln⁡∣u∣/2\int (1/u) du/2 = \ln|u|/2.

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