Q.1 What is the integral of sin(x) with respect to x?
-cos(x)
cos(x)
sin(x)
-sin(x)
Explanation - The integral of sin(x) dx is -cos(x) + C.
Correct answer is: -cos(x)
Q.2 Which of the following is the formula for Simpson's 1/3 rule?
∫f(x)dx ≈ h/3 [f(x0)+4f(x1)+f(x2)]
∫f(x)dx ≈ h/2 [f(x0)+f(x1)]
∫f(x)dx ≈ h [f(x0)+f(x1)+f(x2)+f(x3)]
∫f(x)dx ≈ h/4 [f(x0)+3f(x1)+3f(x2)+f(x3)]
Explanation - Simpson's 1/3 rule approximates the integral using a quadratic polynomial, applicable for an even number of subintervals.
Correct answer is: ∫f(x)dx ≈ h/3 [f(x0)+4f(x1)+f(x2)]
Q.3 The integral ∫e^(2x) dx is equal to:
1/2 e^(2x)
2 e^(2x)
e^(2x)
ln|2x|
Explanation - The integral of e^(ax) dx is (1/a)e^(ax) + C, so here a = 2.
Correct answer is: 1/2 e^(2x)
Q.4 Which numerical method uses trapezoids to approximate the area under a curve?
Trapezoidal Rule
Simpson's Rule
Midpoint Rule
Euler's Method
Explanation - The Trapezoidal Rule approximates the integral by dividing the area under the curve into trapezoids.
Correct answer is: Trapezoidal Rule
Q.5 If ∫(1/x) dx = ?
ln|x| + C
1/x + C
e^x + C
x + C
Explanation - The integral of 1/x dx is the natural logarithm of the absolute value of x.
Correct answer is: ln|x| + C
Q.6 Which method is best for approximating integrals when the function is tabulated?
Trapezoidal Rule
Integration by Parts
Substitution Method
Partial Fraction
Explanation - Trapezoidal Rule works well for tabulated data because it only requires function values at discrete points.
Correct answer is: Trapezoidal Rule
Q.7 The integral of cos(x) dx is:
sin(x)
-sin(x)
cos(x)
-cos(x)
Explanation - The derivative of sin(x) is cos(x), hence ∫cos(x) dx = sin(x) + C.
Correct answer is: sin(x)
Q.8 What is the value of ∫0^1 x^2 dx?
1/3
1/2
1
2/3
Explanation - The integral of x^n dx is x^(n+1)/(n+1). Here n = 2, so ∫0^1 x^2 dx = 1^3/3 - 0 = 1/3.
Correct answer is: 1/3
Q.9 In numerical integration, the Midpoint Rule estimates the integral by:
Using the function value at the midpoint of each subinterval
Using trapezoids under the curve
Fitting a quadratic through points
Using the first derivative of the function
Explanation - The Midpoint Rule approximates the integral by summing the area of rectangles whose height is the function value at the midpoint of each subinterval.
Correct answer is: Using the function value at the midpoint of each subinterval
Q.10 The integral ∫(x^3 - 2x) dx equals:
x^4/4 - x^2 + C
3x^2 - 2x + C
x^4 - x^2 + C
x^3/3 - 2x + C
Explanation - Integrate term by term: ∫x^3 dx = x^4/4, ∫-2x dx = -x^2.
Correct answer is: x^4/4 - x^2 + C
Q.11 Which method is used to approximate ∫f(x)dx when f(x) is difficult to integrate analytically?
Numerical Integration
Partial Fractions
Substitution
Integration by Parts
Explanation - Numerical methods like Trapezoidal or Simpson's rule are used when analytical integration is hard or impossible.
Correct answer is: Numerical Integration
Q.12 The integral ∫(1/(1+x^2)) dx is:
arctan(x) + C
ln|1+x^2| + C
x + C
1/(1+x) + C
Explanation - The derivative of arctan(x) is 1/(1+x^2), so the integral is arctan(x) + C.
Correct answer is: arctan(x) + C
Q.13 Which method approximates the integral using parabolic arcs?
Simpson's Rule
Trapezoidal Rule
Midpoint Rule
Euler's Method
Explanation - Simpson's Rule approximates the area under the curve using second-degree polynomials (parabolas).
Correct answer is: Simpson's Rule
Q.14 The definite integral ∫1^2 (2x + 3) dx equals:
5
6
7
8
Explanation - ∫(2x+3) dx = x^2 + 3x, evaluate from 1 to 2: (4+6)-(1+3)=7.
Correct answer is: 7
Q.15 Which of the following is an example of improper integral?
∫1^∞ 1/x^2 dx
∫0^1 x^2 dx
∫0^π sin(x) dx
∫1^2 (x+1) dx
Explanation - An improper integral has either infinite limits or an integrand with singularities within the interval.
Correct answer is: ∫1^∞ 1/x^2 dx
Q.16 Which method divides the integration interval into equal subintervals and applies trapezoids?
Trapezoidal Rule
Simpson's Rule
Midpoint Rule
Integration by Parts
Explanation - The Trapezoidal Rule divides the interval into n subintervals and approximates each area with a trapezoid.
Correct answer is: Trapezoidal Rule
Q.17 The integral ∫(x^2 + 1) dx is equal to:
x^3/3 + x + C
x^2 + ln|x| + C
2x + 1 + C
x^3 + x + C
Explanation - Integrate each term: ∫x^2 dx = x^3/3, ∫1 dx = x.
Correct answer is: x^3/3 + x + C
Q.18 Which method of numerical integration is exact for polynomials up to degree 3?
Simpson's 3/8 Rule
Simpson's 1/3 Rule
Trapezoidal Rule
Midpoint Rule
Explanation - Simpson's 3/8 Rule can exactly integrate cubic polynomials, unlike Simpson's 1/3 rule which is exact for quadratics.
Correct answer is: Simpson's 3/8 Rule
Q.19 The integral ∫x e^x dx can be solved using:
Integration by Parts
Substitution
Partial Fractions
Trapezoidal Rule
Explanation - Integration by parts formula ∫u dv = uv - ∫v du is suitable when the integrand is a product of two functions like x and e^x.
Correct answer is: Integration by Parts
Q.20 For ∫0^1 (4 - x^2) dx, the exact value is:
11/3
8/3
5/3
7/3
Explanation - ∫(4 - x^2) dx = 4x - x^3/3; evaluate 0 to 1: 4 - 1/3 = 11/3.
Correct answer is: 11/3
Q.21 In numerical methods, the error in Trapezoidal Rule is proportional to:
h^2
h^3
h
h^4
Explanation - The error in the Trapezoidal Rule is proportional to the square of the step size h and the second derivative of the function.
Correct answer is: h^2
Q.22 Which substitution is appropriate for ∫sin^2(x) dx?
Use sin^2(x) = (1 - cos(2x))/2
Use sin^2(x) = 1 - sin(x)
Use x = tan(θ)
Use u = e^x
Explanation - The power-reduction formula transforms sin^2(x) into a form easily integrable.
Correct answer is: Use sin^2(x) = (1 - cos(2x))/2
Q.23 The integral ∫0^π sin(x) dx equals:
2
1
0
π
Explanation - ∫0^π sin(x) dx = [-cos(x)]0^π = -cos(π) + cos(0) = 2.
Correct answer is: 2
Q.24 In Simpson's 1/3 Rule, the number of subintervals must be:
Even
Odd
Prime
Multiple of 3
Explanation - Simpson's 1/3 Rule requires an even number of subintervals to apply the alternating 4-2 coefficients correctly.
Correct answer is: Even
Q.25 The integral ∫(cos^2(x)) dx can be computed by:
Using cos^2(x) = (1 + cos(2x))/2
Using cos^2(x) = 1 - sin^2(x)
Direct integration
Integration by parts
Explanation - The power-reduction formula allows integration of cos^2(x) easily.
Correct answer is: Using cos^2(x) = (1 + cos(2x))/2