Integrals of Piecewise Functions

JEE Main Mathematics · Definite Integration · 23 questions, latest first

• Let f(x)= {1/3, x ≤ π / 2; (b(1- sin x))/((π-2 x)^2), x>π / 2}. If f is continuous at x=π / 2, then the value of ∫_0^3 b-6|x^2+2…20268 April, Shift 2 · Q19
• Let ∫_-2^2(| sin x|+[x sin x]) d x=2(3- cos 2)+β, where [·] is the greatest integer function. Then β sin ((β)/2) equals:20264 April, Shift 1 · Q19
• Let [·] denote the greatest integer function. Then the value of ∫_0^3((e^x+e^-x)/([x]!)) d x is:20262 April, Shift 1 · Q19
• Let [·] denote the greatest integer function. Then ∫_-(π)/2^((π)/2)((12(3+[x]))/(3+[ sin x]+[ cos x])) d x is equal to:202628 January, Shift 2 · Q17
• The value of ∫_-(π)/2^((π)/2)(1/([x]+4)) d x, where [·] denotes the greatest integer function, is202622 January, Shift 1 · Q20
• 6 ∫_0^π|( sin 3 x+ sin 2 x+ sin x)| d x is equal to ____.202621 January, Shift 1 · Q25
• The integral ∫_-1^(3/2)(|π^2 x sin (π x)|) d x is equal to:20258 April, Shift 2 · Q19
• Let the domain of the function f(x)= log_2 log_4 log_6(3+4 x-x^2) be (a, b). If ∫_0^b-a[x^2] d x=p-√q-√r, p, q, r ∈ N, gcd(p, q,…20253 April, Shift 1 · Q19
• Let [·] denote the greatest integer function. If ∫_0^e^3[1/(e^x-1)] d x=α- log_e 2, then α^3 is equal to ____.20252 April, Shift 1 · Q23
• If 24 ∫_0^((π)/4)( sin |4 x-(π)/12|+[2 sin x]) d x=2 π+α, where [·] denotes the greatest integer function, then α is equal to…202529 January, Shift 2 · Q25
• Let [t] denote the largest integer less than or equal to t. If ∫_0^3([x^2]+[x^2/2]) d x=a+b √2-√3-√5+c √6-√7, where a, b, c ∈ Z,…20246 April, Shift 2 · Q25
• Let f(x)= {-2, -2 ≤ x ≤ 0; x-2, 0<x ≤ 2} and h(x)=f(|x|)+|f(x)|. Then ∫_-2^2 h(x) d x is equal to:20244 April, Shift 1 · Q10
• The value of 9 ∫_0^9[√((10 x)/(x+1))] d x, where [t] denotes the greatest integer less than or equal to t, is ____.202430 January, Shift 1 · Q25
• If ∫_((π)/6)^((π)/3) √(1- sin 2 x) d x=α+β √2+γ √3, where α, β and γ are rational numbers, then 3 α+4 β-γ is equal to ____.202429 January, Shift 2 · Q26
• If ∫_-0.15^0.15|100 x^2-1| d x=k/3000, then k is equal to ____.202312 April, Shift 1 · Q26
• Let the function f:[0,2] → R be defined as f(x)= {e^min {x^2, x-[x]}, x ∈[0,1); e^[x- log_e x], x ∈[1,2]} where [t] denotes the…202311 April, Shift 2 · Q11
• Let [t] denote the greatest integer function. If ∫_0^2.4[x^2] d x=α+β √2+γ √3+δ √5, then α+β+γ+δ is equal to ____.20238 April, Shift 2 · Q26
• Let [t] denote the greatest integer ≤ t. Then 2/(π) ∫_π / 6^5 π / 6(8[cosec x]-5[ cot x]) d x is equal to ____.20238 April, Shift 1 · Q24
• If [t] denotes the greatest integer ≤ t, then the value of (3(e-1))/e ∫_1^2 x^2 e^[x]+[x^3] d x is:202330 January, Shift 1 · Q69
• Let [x] denote the greatest integer ≤ x. Consider the function f(x)= max {x^2, 1+[x]}. Then the value of the integral ∫_0^2 f(x)…202329 January, Shift 1 · Q74
• If ∫_(1/3)^3| log_e x| d x=m/n log_e(n^2/e), where m and n are coprime natural numbers, then m^2+n^2-5 is equal to ____202325 January, Shift 2 · Q86
• The minimum value of the function f(x)=∫_0^2 e^|x-t| d t is:202325 January, Shift 1 · Q70
• The value of 12 ∫_0^3|x^2-3 x+2| d x is ____202324 January, Shift 1 · Q87