← Back to Mathematics Chapters
Ch-7: Integrals
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. Integration is essentially the reverse process of:
A. Differentiation
B. Multiplication
C. Addition
D. Taking limits
Show Answer
Answer: A. Differentiation
Explanation: Integration finds a function whose derivative is the given function, making it the inverse operation of differentiation.
2. The integral ∫xⁿ dx (for n ≠ -1) is equal to:
A. xⁿ⁺¹ / (n + 1) + C
B. n xⁿ⁻¹
C. xⁿ / n
D. n! xⁿ
Show Answer
Answer: A. xⁿ⁺¹ / (n + 1) + C
Explanation: This is the standard power rule for integration, valid for all n except n = -1.
3. The integral ∫(1/x) dx is equal to:
A. ln|x| + C
B. 1/x² + C
C. x + C
D. -1/x² + C
Show Answer
Answer: A. ln|x| + C
Explanation: This special case, excluded from the power rule, integrates to the natural logarithm of the absolute value of x.
4. The integral ∫eˣ dx is equal to:
A. eˣ + C
B. eˣ/x + C
C. x eˣ + C
D. ln(eˣ)
Show Answer
Answer: A. eˣ + C
Explanation: The exponential function eˣ is its own antiderivative, just as it is its own derivative.
5. The integration by parts formula is given by:
A. ∫u dv = uv - ∫v du
B. ∫u dv = uv + ∫v du
C. ∫u dv = u + v
D. ∫u dv = u · v always
Show Answer
Answer: A. ∫u dv = uv - ∫v du
Explanation: This formula is derived from the product rule of differentiation and is used to integrate products of functions.
6. The method of substitution is used to simplify an integral by:
A. Changing the variable of integration
B. Differentiating the integrand first
C. Multiplying the integrand by a constant
D. Simply adding a constant
Show Answer
Answer: A. Changing the variable of integration
Explanation: Substitution transforms a complicated integral into a simpler one by introducing a new variable related to part of the integrand.
7. A definite integral ∫[a to b] f(x) dx represents:
A. The signed area under the curve between x = a and x = b
B. The slope of the curve
C. The derivative of f at x = a
D. The value of f at x = b
Show Answer
Answer: A. The signed area under the curve between x = a and x = b
Explanation: Geometrically, a definite integral gives the net signed area bounded by the curve and the x-axis over the given interval.
8. Partial fractions are primarily used to integrate:
A. Rational functions with factorable denominators
B. Exponential functions only
C. Trigonometric functions only
D. Polynomial functions only
Show Answer
Answer: A. Rational functions with factorable denominators
Explanation: Breaking a complex rational expression into simpler fractions makes each term easier to integrate individually.
9. The Fundamental Theorem of Calculus establishes the connection between:
A. Differentiation and integration
B. Limits and continuity
C. Matrices and determinants
D. Vectors and scalars
Show Answer
Answer: A. Differentiation and integration
Explanation: This theorem shows that differentiation and integration are essentially inverse operations of one another.
10. The integral ∫cos x dx is equal to:
A. sin x + C
B. -sin x + C
C. cos x + C
D. -cos x + C
Show Answer
Answer: A. sin x + C
Explanation: This is a standard result following directly from d/dx(sin x) = cos x.
11. The integral ∫sec²x dx is equal to:
A. tan x + C
B. sec x + C
C. cot x + C
D. -tan x + C
Show Answer
Answer: A. tan x + C
Explanation: This follows directly from the derivative rule d/dx(tan x) = sec²x.
12. A definite integral with identical upper and lower limits, ∫[a to a] f(x) dx, always equals:
A. Zero
B. One
C. Infinity
D. It is always undefined
Show Answer
Answer: A. Zero
Explanation: When the upper and lower limits coincide, the interval of integration has zero width, giving a value of zero.
