Integration with Inverse Trig Functions Practice

Sample questions

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1. Sample 1difficulty 2/5

   $\displaystyle\int_0^1 \dfrac{1}{1+x^2}\,dx =$

   • A

     $0$

   • B

     $\tfrac{\pi}{4}$

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   • C

     $\pi$

   • D

     $\tfrac{\pi}{2}$

   Why

   $[\arctan x]_0^1 = \pi/4 - 0 = \pi/4$.

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2. Sample 2difficulty 2/5

   $\displaystyle\int \dfrac{1}{\sqrt{1-x^2}}\,dx =$

   • A

     $\ln|1-x^2| + C$

   • B

     $\arcsin x + C$

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   • C

     $\arctan x + C$

   • D

     $\arccos x + C$

   Why

   $\arcsin'(x) = 1/\sqrt{1 - x^2}$.

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3. Sample 3difficulty 2/5

   $\displaystyle\int \dfrac{1}{1 + x^2}\,dx =$

   • A

     $\arcsin x + C$

   • B

     $\arctan x + C$

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   • C

     $\ln(1 + x^2) + C$

   • D

     $\dfrac{x^2}{1+x^2} + C$

   Why

   $\arctan'(x) = 1/(1+x^2)$.

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4. Sample 4difficulty 3/5

   $\displaystyle\int \dfrac{1}{4 + x^2}\,dx =$

   • A

     $\dfrac{1}{4 + x^2} + C$

   • B

     $\dfrac{1}{2}\arctan(x/2) + C$

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   • C

     $\dfrac{1}{4}\arctan x + C$

   • D

     $\arctan(x/2) + C$

   Why

   Pattern $1/(a^2 + x^2)$: antiderivative $(1/a)\arctan(x/a)$.

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