Properties of Definite Integrals

JEE Main Mathematics · Definite Integration · 28 questions, latest first

• The value of the integral ∫_-(π)/4^((π)/4)((32 cos^4 x)/(1+e^sin x)) d x is:20266 April, Shift 1 · Q18
• The value of the integral ∫_((π)/24)^((5 π)/24) (d x)/(1+√[3](tan 2 x)) is:202623 January, Shift 1 · Q19
• If ∫_0^1 4 cot^-1(1-2 x+4 x^2) d x=a tan^-1(2)-b log_e(5), where a, b ∈ N, then (2 a+b) is equal to ____.202621 January, Shift 2 · Q25
• Let f(x) be a positive function and I_1=∫_-1/2^1 2 x f(2 x(1-2 x)) d x and I_2=∫_-1^2 f(x(1-x)) d x. Then the value of I_2/I_1 is…20258 April, Shift 2 · Q18
• The integral ∫_0^π ((x+3) sin x)/(1+3 cos^2 x) d x is equal to20257 April, Shift 1 · Q18
• The value of ∫_-1^1 ((1+√(|x|-x)) e^x+(√(|x|-x)) e^-x)/(e^x+e^-x) d x is equal to20254 April, Shift 1 · Q19
• The integral ∫_0^π (8 x d x)/(4 cos^2 x+ sin^2 x) is equal to20253 April, Shift 2 · Q18
• Let (a, b) be the point of intersection of the curve x^2=2 y and the straight line y-2 x-6=0 in the second quadrant. Then the…20252 April, Shift 2 · Q19
• Let f: R → R be a twice differentiable function such that f(2)=1. If F(x)=x f(x) for all x ∈ R, ∫_0^2 x F^′(x) d x=6 and ∫_0^2…202528 January, Shift 2 · Q17
• If ∫_-(π)/2^((π)/2) (96 x^2 cos^2 x)/((1+e^x)) d x=π(α π^2+β), α, β ∈ Z, then (α+β)^2 equals202528 January, Shift 1 · Q20
• If I=∫_0^((π)/2) (sin^(3/2) x)/(sin^(3/2) x+ cos^(3/2) x) d x, then ∫_0^2 I (x sin x cos x)/(sin^4 x+ cos^4 x) d x equals:202523 January, Shift 2 · Q18
• The value of ∫_e^2^e^4 1/x((e^(( log_e x)^2+1)^-1)/(e^(( log_e x)^2+1)^-1+e^((6- log_e x)^2+1)^-1)) d x is202523 January, Shift 1 · Q19
• If f(t)=∫_0^π (2 x d x)/(1- cos^2 t sin^2 x), 0<t<π, then the value of ∫_0^((π)/2) (π^2 d t)/(f(t)) equals ____.20245 April, Shift 2 · Q25
• The value of ∫_-π^π (2 y(1+ sin y))/(1+ cos^2 y) d y is:20245 April, Shift 1 · Q10
• If the value of the integral ∫_-1^1 (cos α x)/(1+3^x) d x is 2/(π). Then, a value of α is20244 April, Shift 2 · Q10
• The value of ∫_0^1(2 x^3-3 x^2-x+1)^(1/3) d x is equal to:20241 February, Shift 2 · Q11
• The value of the intergral ∫_0^π / 4 (x d x)/(sin^4(2 x)+ cos^4(2 x)) equals:20241 February, Shift 1 · Q10
• If ∫_-π / 2^π / 2 (8 √2 cos x d x)/((1+e^sin x)(1+ sin^4 x))=α π+β log_e(3+2 √2), where α, β are integers, then α^2+β^2 equals…20241 February, Shift 1 · Q27
• |120/(π^3) ∫_0^π (x^2 sin x cos x)/(sin^4 x+ cos^4 x) d x| is equal to ____.202431 January, Shift 2 · Q26
• Let f: R → R be a function defined by f(x)=4^x/(4^x+2) and M=∫_f(a)^f(1-a) x sin^4(x(1-x)) d x, N=∫_f(a)^f(1-a) sin^4(x(1-x)) d…202431 January, Shift 1 · Q26
• If the value of the integral ∫_-(π)/2^((π)/2)((x^2 cos x)/(1+π^x)+(1+ sin^2 x)/(1+e^sin x^2023)) d x=(π)/4(π+a)-2, then the value…202429 January, Shift 1 · Q11
• If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1), and I_1=∫_a^b x sin (4 x-x^2) d x,…202427 January, Shift 1 · Q12
• If ∫_0^1 1/((5+2 x-2 x^2)(1+e^(2-4 x))) d x=1/(α) log_e((α+1)/(β)), α, β>0, then α^4-β^4 is equal to202315 April, Shift 1 · Q10
• If f: R → R be a continuous function satisfying ∫_0^((π)/2) f( sin 2 x) sin x d x+α ∫_0^((π)/4) f( cos 2 x) cos x d x=0, then the…202311 April, Shift 2 · Q12
• Let f(x) be a function satisfying f(x)+f(π-x)=π^2, ∀ x ∈ R. Then ∫_0^π f(x) sin x d x is equal to20236 April, Shift 2 · Q11
• If ∫_0^π (5^cos x(1+ cos x cos 3 x+ cos^2 x+ cos^3 x cos 3 x) d x)/(1+5^cos x)=(k π)/16, then k is equal to ____.20231 February, Shift 2 · Q86
• The value of the integral ∫_1 / 2^2 (tan^-1 x)/x d x is equal to202329 January, Shift 2 · Q68
• The value of 8/(π) ∫_0^((π)/2) (( cos x)^2023)/(( sin x)^2023+( cos x)^2023) d x is ____202324 January, Shift 1 · Q88