Integrals with inverse trig, long division and completing the square

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   Ari substitutes and uses an inverse trig rule to find ∫ eˣ/(1 + e^(2x)) dx. Which is it?

   • ln(1 + e^(2x)) + C
   • arctan(eˣ) + C
   • arctan(e^(2x)) + C
2. 2.

   Bo substitutes and uses an inverse trig rule to find ∫ x/(1 + x⁴) dx. Which is it?

   • (1/2) arctan(x⁴) + C
   • (1/2) arctan(x²) + C
   • (1/4) ln(1 + x⁴) + C
3. 3.

   Esme substitutes and uses an inverse trig rule to find ∫ cos x/(1 + sin² x) dx. Which is it?

   • arctan(sin x) + C
   • ln(1 + sin² x) + C
   • arctan(cos x) + C
4. 4.

   Jun substitutes and uses an inverse trig rule to find ∫ 1/(x√(1 − (ln x)²)) dx. Which is it?

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

   Kira divides first to evaluate ∫ from 1 to 4 of (x² + 3)/(x + 1) dx. What is it, to the nearest hundredth?

   8.17
6. 6.

   Luz divides first to evaluate ∫ from 0 to 3 of (x² + 5)/(x + 2) dx. What is it, to the nearest hundredth?

   6.75
7. 7.

   Milo divides first to evaluate ∫ from 1 to 3 of (2x² + 1)/(x + 1) dx. What is it, to the nearest hundredth?

   6.08
8. 8.

   Noah divides first to evaluate ∫ from 0 to 1 of (x³ + 2)/(x + 1) dx. What is it, to the nearest hundredth?

   1.53
9. 9.

   Pia decides between a log and an arctan for ∫ x/(x² + 9) dx. Which is the antiderivative?

   • ln(x² + 9) + C
   • (1/2) ln(x² + 9) + C
   • (1/3) arctan(x/3) + C
10. 10.

    Zee decides between a log and an arctan for ∫ 3/(x² + 4) dx. Which is the antiderivative?

    • (3/2) arctan(x/2) + C
    • 3 arctan(x/2) + C
    • (3/2) ln(x² + 4) + C