Integrals

33 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Integrals' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Integrals'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Integrals'? medium MCQ Problem Solving & Optimization
• 4 Find the area (in square units) enclosed between the curves y = x^2 and y = 2x+3. medium 3 marks area between curves using definite integrals
• 5 Evaluate the integral ∫ (3x^2+2x) dx and choose the correct antiderivative. easy MCQ 1 mark basic integration
• 6 Determine for which values of the parameter p the improper integral ∫_1^(∞)dx/x^p converges, and give its value when it converges. hard 4 marks convergence of improper integrals
• 7 Find the value of the definite integral ∫_0^(π/4)tan x dx. medium 2 marks definite integral of trigonometric function
• 8 Find the antiderivative of the function 4x^3-5x^2+2 and write the general form of the integral ∫ (4x^3-5x^2+2),dx. easy 1 mark Basic antiderivative
• 9 Find the value of the improper integral ∫_0^(∞)x^3e^-x dx using integration by parts. hard 4 marks Improper integral, Gamma function
• 10 Evaluate the definite integral ∫_0^2(x^2+1)/(x+1)^2 dx using algebraic manipulation and substitution. hard 4 marks Definite integral, algebraic manipulation
• 11 The curves y=4x^2 and y=8x+8 intersect at two points. The area of the region bounded by these two curves is given by ∫_a^b(8x+8-4x^2) dx. Which of the following options gives the correct value of this area? medium MCQ 2 marks Area between curves
• 12 Find the area enclosed between the curves y=x^2 and y=4x. medium 2 marks Area between curves
• 13 Find the general antiderivative of the function f(x)=5x^4-3x^2+7 and state the correct antiderivative among the following options: easy MCQ 1 mark Basic antiderivative using power rule
• 14 Determine for which values of the parameter a the improper integral ∫_0^1x^-aln x dx converges, and give its value when it converges. hard 4 marks Convergence of logarithmic improper integrals
• 15 Evaluate the improper integral ∫_0^(∞)x^2e^-3x dx using integration by parts or the gamma function. hard 3 marks Improper integrals with exponential decay
• 16 Evaluate the improper integral displaystyle int_0^infty x^4e^-2x,dx using the gamma function or standard formulas. hard 4 marks Gamma function and exponential integrals
• 17 Evaluate the improper integral displaystyle int_0^1 x^2ln x,dx. medium 3 marks Improper integral involving logarithm
• 18 Find the area enclosed between the curves y=x^3 and y=3x^2 between their intersection points. medium 3 marks Area between curves
• 19 Evaluate the improper integral ∫_0^(∞) x^2.5 e^-x dx and give the numerical value up to three decimal places. hard 4 marks Gamma function evaluation of improper integrals
• 20 Evaluate the definite integral ∫_0^1 (3x^2-2x+1) dx. easy 2 marks Definite integral evaluation
• 21 Determine all real values of the parameter a for which the improper integral ∫_0^(∞) (ln x)/(1+x^2)^a dx converges. For those values, evaluate the integral when a=2. hard 5 marks Convergence of improper integrals with parameters
• 22 Find the value of the definite integral ∫_0^3 (2x-1),dx. easy 1 mark Definite integral of a polynomial
• 23 Find the area enclosed between the curves y=x^3 and y=4x^2-3x between their intersection points. hard 4 marks Area between curves with multiple intervals
• 24 For which values of the real parameter a does the improper integral ∫_0^1 x^aln x dx converge? When it converges, find its value. hard 3 marks Improper integral with parameter
• 25 Evaluate the improper integral ∫_0^(∞) x e^-2x,dx using the gamma function or integration by parts. medium 2 marks Improper integral with exponential decay
• 26 The curves y = x^2 and y = 4x - 3 intersect at two points. What is the area of the region bounded by these curves? medium MCQ 3 marks Area between two curves
• 27 Evaluate the improper integral I=∫_0^(∞) x^(3/2) e^-x dx and give the numerical value correct to three decimal places. hard 4 marks Gamma function for improper integrals
• 28 Find the antiderivative F(x) of f(x)=6x^5-4x^3+9 and then evaluate the definite integral ∫_1^3 f(x) dx. easy 2 marks Finding antiderivative and evaluating definite integral
• 29 Determine all real values of the parameter p for which the improper integral ∫_1^(∞) (ln x)/x^p dx converges. Then evaluate the integral when p=3. hard 5 marks Convergence of improper integrals with parameters and integration by parts
• 30 Evaluate the integral I=∫_0^(π/2)sin^2x dx. medium MCQ 3 marks Definite integral using trigonometric identities
• 31 Find the antiderivative F(x) of the function f(x)=3x^4-2x+5. Then compute the definite integral ∫_0^2 f(x) dx. easy 2 marks Antiderivative and definite integral
• 32 Find the area enclosed between the curves y = x^3 and y = 2x^2 - x. Give the exact value and its decimal approximation up to three decimal places. medium 4 marks Area between two curves
• 33 Consider the improper integral I(a,b)=∫_0^(∞) x^a e^(-b x) dx, where a>-1 and b>0. hard 5 marks Gamma function and convergence of improper integrals