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Quiz: Transcendental Integrals

Test your understanding of transcendental integrals with these review questions.

1. What is ∫eˣ dx?

  1. xeˣ + C
  2. eˣ + C
  3. eˣ/x + C
  4. eˣ⁻¹ + C
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The correct answer is B. Since d/dx[eˣ] = eˣ, the antiderivative of eˣ is itself: ∫eˣ dx = eˣ + C.

Concept Tested: Integral of e to x

2. What is ∫1/x dx?

  1. x⁻¹ + C
  2. ln(x) + C
  3. ln|x| + C
  4. −1/x² + C
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The correct answer is C. The integral of 1/x is ln|x| + C. The absolute value is needed because ln is only defined for positive arguments, but 1/x is defined for x ≠ 0.

Concept Tested: Integral of 1 Over x

3. What is ∫2ˣ dx?

  1. 2ˣ + C
  2. 2ˣ/ln(2) + C
  3. x·2ˣ⁻¹ + C
  4. 2ˣ·ln(2) + C
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The correct answer is B. For ∫aˣ dx = aˣ/ln(a) + C. So ∫2ˣ dx = 2ˣ/ln(2) + C. This comes from the fact that d/dx[aˣ] = aˣ·ln(a).

Concept Tested: Integral of a to x

4. What is ∫1/(1 + x²) dx?

  1. ln(1 + x²) + C
  2. arctan(x) + C
  3. arcsin(x) + C
  4. 1/2 · ln(1 + x²) + C
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The correct answer is B. Since d/dx[arctan(x)] = 1/(1 + x²), we have ∫1/(1 + x²) dx = arctan(x) + C.

Concept Tested: Arctan Integral

5. What is ∫1/√(1 − x²) dx?

  1. arctan(x) + C
  2. arcsin(x) + C
  3. arccos(x) + C
  4. ln|x| + C
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The correct answer is B. Since d/dx[arcsin(x)] = 1/√(1 − x²), we have ∫1/√(1 − x²) dx = arcsin(x) + C.

Concept Tested: Arcsin Integral

6. What is ∫sec(x)tan(x) dx?

  1. sec(x) + C
  2. tan(x) + C
  3. sec²(x) + C
  4. −sec(x) + C
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The correct answer is A. Since d/dx[sec(x)] = sec(x)tan(x), the antiderivative of sec(x)tan(x) is sec(x) + C.

Concept Tested: Integral of Sec Tan

7. What is ∫csc²(x) dx?

  1. cot(x) + C
  2. −cot(x) + C
  3. csc(x) + C
  4. −csc(x)cot(x) + C
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The correct answer is B. Since d/dx[−cot(x)] = csc²(x), we have ∫csc²(x) dx = −cot(x) + C.

Concept Tested: Integral of Csc Squared

8. What is ∫e^(3x) dx?

  1. e^(3x) + C
  2. 3e^(3x) + C
  3. e^(3x)/3 + C
  4. e^(3x+1) + C
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The correct answer is C. Using substitution (or recognizing the pattern): ∫e^(3x) dx = e^(3x)/3 + C. Check: d/dx[e^(3x)/3] = e^(3x).

Concept Tested: Integral of e to x

9. The integral ∫1/x dx = ln|x| + C uses absolute value because:

  1. It makes the answer positive
  2. ln is only defined for positive numbers, but 1/x is defined for x ≠ 0
  3. It's a convention that doesn't matter
  4. The derivative requires it
Show Answer

The correct answer is B. The function 1/x is defined for all x ≠ 0, but ln(x) is only defined for x > 0. Using |x| extends the antiderivative to negative x values.

Concept Tested: Natural Log Integral

10. What is ∫csc(x)cot(x) dx?

  1. csc(x) + C
  2. −csc(x) + C
  3. cot(x) + C
  4. −cot(x) + C
Show Answer

The correct answer is B. Since d/dx[−csc(x)] = csc(x)cot(x), we have ∫csc(x)cot(x) dx = −csc(x) + C.

Concept Tested: Integral of Csc Cot