Integrals with inverse trig, long division and completing the square

Answer Key - Grade 12 Math Worksheet - Building Confidence

Circle the best answer, or write your answer on the line.

1. 1.

   Ari finds ∫ 5/(1 + x²) dx using an inverse trig rule. Which is it?

   • 5 arcsin x + C
   • 5 ln(1 + x²) + C
   • 5 arctan x + C
2. 2.

   Bo finds ∫ 3/√(1 − x²) dx using an inverse trig rule. Which is it?

   • 3 arcsin x + C
   • 3 arctan x + C
   • 3 ln(1 − x²) + C
3. 3.

   Esme finds ∫ 2/(9 + x²) dx using an inverse trig rule. Which is it?

   • (2/3) arctan(x/3) + C
   • (2/3) arcsin(x/3) + C
   • 2 arctan(x/3) + C
4. 4.

   Jun finds ∫ 4/(16 + x²) dx using an inverse trig rule. Which is it?

   • arcsin(x/4) + C
   • arctan(x/4) + C
   • 4 arctan(x/4) + C
5. 5.

   Luz divides first: (x² + 5)/x = x + 5/x. What is ∫ from 1 to 3 of (x² + 5)/x dx, to the nearest hundredth?

   9.49
6. 6.

   Milo divides first: (x² + 2)/x = x + 2/x. What is ∫ from 1 to 4 of (x² + 2)/x dx, to the nearest hundredth?

   10.27
7. 7.

   Noah divides first: (x² + 6)/x = x + 6/x. What is ∫ from 1 to 2 of (x² + 6)/x dx, to the nearest hundredth?

   5.66
8. 8.

   Pia completes the square in x² + 2x + 5 to find ∫ 1/(x² + 2x + 5) dx. Which is it?

   • (1/4) arctan((x + 1)/2) + C
   • (1/2) arctan((x + 1)/2) + C
   • (1/2) arctan((x − 1)/2) + C
   • arctan((x + 1)/2) + C
9. 9.

   Ren completes the square in x² − 4x + 13 to find ∫ 1/(x² − 4x + 13) dx. Which is it?

   • (1/3) arctan((x + 2)/3) + C
   • (1/9) arctan((x − 2)/3) + C
   • (1/3) arctan((x − 2)/3) + C
   • arctan((x − 2)/3) + C
10. 10.

    Tala completes the square in x² + 6x + 25 to find ∫ 1/(x² + 6x + 25) dx. Which is it?

    • arctan((x + 3)/4) + C
    • (1/4) arctan((x − 3)/4) + C
    • (1/4) arctan((x + 3)/4) + C
    • (1/16) arctan((x + 3)/4) + C