Chapter Exam

Chapter 7: Integrals — Online MCQ Test

Q: Integration is the reverse process of which mathematical operation?

Answer: A. Differentiation — Integration is defined as the inverse or anti-derivative process of differentiation, making it the exact reverse operation.

Q: What is the arbitrary constant added to an indefinite integral called?

Answer: B. Constant of integration — The arbitrary constant C added to indefinite integrals accounts for the family of curves differing by a constant.

Q: Using integration by substitution, ∫ sin(2x) dx = ?

Answer: A. -cos(2x)/2 + C — Let u = 2x, du = 2dx. Then ∫ sin(2x) dx = (1/2)∫ sin(u) du = -cos(u)/2 + C = -cos(2x)/2 + C.

Q: Consider ∫ (3x+5)/(x²+4x+8) dx. After completing the square in denominator and substitution, this becomes:

Answer: A. An integral of form ∫ (du/u) + ∫ (dv/(v²+a²)) — The numerator 3x+5 can be split: (3/2)(2x+4) + (-1). The first part gives a logarithmic integral, the second gives a tan⁻¹ integral after completing the square.

MATHS · CLASS 12 SECOND PUC · Karnataka State Board

Practice Chapter 7: Integrals with a free chapter-wise online MCQ test for Karnataka State Board CLASS 12 SECOND PUC MATHS. This chapter covers: Inverse Process: Integration acts as the exact reverse process of differentiation, earning it the alternative name "anti-derivative."Indefinite Integrals: Indefinite integration ma.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 7: Integrals — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 12 SECOND PUC MATHS — Chapter 7: Integrals, with answers and explanations. These are sample questions; the exam has its own separate question set.

Integration is the reverse process of which mathematical operation?

A. Differentiation ✓
B. Multiplication
C. Division
D. Exponentiation

Answer: A. Differentiation
Integration is defined as the inverse or anti-derivative process of differentiation, making it the exact reverse operation.

What is the arbitrary constant added to an indefinite integral called?

A. Integration factor
B. Constant of integration ✓
C. Differential constant
D. Limit constant

Answer: B. Constant of integration
The arbitrary constant C added to indefinite integrals accounts for the family of curves differing by a constant.

What is the fundamental difference between definite and indefinite integrals?

A. Indefinite integrals have limits; definite integrals do not
B. Definite integrals have limits and produce a scalar value; indefinite integrals do not have limits and include C ✓
C. Both are identical in form and calculation
D. Definite integrals are always larger than indefinite integrals

Answer: B. Definite integrals have limits and produce a scalar value; indefinite integrals do not have limits and include C
Definite integrals ∫ₐᵇ f(x)dx have numerical limits and result in a specific value, while indefinite integrals produce a family of functions with constant C.

Using integration by substitution, ∫ sin(2x) dx = ?

A. -cos(2x)/2 + C ✓
B. -cos(2x) + C
C. cos(2x)/2 + C
D. 2cos(2x) + C

Answer: A. -cos(2x)/2 + C
Let u = 2x, du = 2dx. Then ∫ sin(2x) dx = (1/2)∫ sin(u) du = -cos(u)/2 + C = -cos(2x)/2 + C.

Consider ∫ (3x+5)/(x²+4x+8) dx. After completing the square in denominator and substitution, this becomes:

A. An integral of form ∫ (du/u) + ∫ (dv/(v²+a²)) ✓
B. A simple logarithmic integral
C. Only a tan⁻¹ integral
D. Impossible to evaluate

Answer: A. An integral of form ∫ (du/u) + ∫ (dv/(v²+a²))
The numerator 3x+5 can be split: (3/2)(2x+4) + (-1). The first part gives a logarithmic integral, the second gives a tan⁻¹ integral after completing the square.