Definite Integration questions
424 published questions with worked solutions.
- Let β(m, n) = _0^1x^ m-1 (1 - x)^ n-1 dx, m, n > 0. If _0^1(1 - x^ 10 )^ 20 dx = a (b, c), then 100(a + b + c) equals _______.
- Let r_k = _0^1(1 - x^7)^kdx _0^1(1 - x^7)^ k+1 dx , k ∈ N. Then the value of _ k=1 ^ 10 1 7(r_k - 1) is equal to _____.
- If f(a + b + 1 – x) = f(x), for all x, where a and b are fixed positive real numbers, then 1 a+b _a^b x(f(x)+f(x+1)) ,dx is equal...
- _ x 0 _0^xt (10t) ,dt x is equal to :
- Let [t] denote the greatest integer less than or equal to t. Then the value of _1^2|2x-[3x]| ,dx is __________.
- Let [t] denote the largest integer less than or equal to t. If _0^3 ([x^2] + [ x^2 2 ] )dx = a + b 2 - 3 - 5 + c 6 - 7, where a,...
- The integral _0^2 ||x-1|-x |dx is equal to___________.
- _ - ^ | -|x| |dx is equal to-
- The value of k ∈ ℕ for which the integral I_n = _0^1(1 - x^k)^ndx, n ∈ ℕ, satisfies 147 I₂₀ = 148 I₂₁ is:
- Let f(x) = |x – 2| and g(x) = f(f(x)), x ∈ [0, 4]. Then _0^3(g(x)-f(x)) ,dx is equal to :
- The value of _ - /2 ^ /2 1 1+e^ x dx is
- If I_1= _0^1(1-x^ 50 )^ 100 dx and I_2= _0^1(1-x^ 50 )^ 101 dx such that I_2= I_1 then equals to -
- Let a,b,c be non-zero real numbers such that _0^1 (1+ ^8x ) (ax^2+bx+c )dx= _0^2 (1+ ^8x ) (ax^2+bx+c )dx. Then the quadratic...
- The value of _0^ 20 ( ^4x + ^4x)dx is equal to:
- Let [·] denote the greatest integer function. Then the value of _0^3 ( e^x + e^ -x [x]! )dx is:
- If = _0^ 2 3 _2(x^2 + 4) ,dx + _2^4 2^x - 4 ,dx, then α² is equal to _______.
- Let _ -2 ^ 2 (| x| + [x x])dx = 2(3- 2) + , where [·] is the greatest integer function. Then ( 2 ) equals:
- If α = 1 and β = 1 + i√2, where i = √−1, are two roots of the equation x³+ax²+bx+c=0, a,b,c ∈ ℝ, then _ -1 ^ 1 (x^3+ax^2+bx+c)dx...
- Let f be a twice differentiable function such that f(x)= _0^x (t-x)dt - _0^x f(t) t ,dt, x∈(−π/2,π/2). Then f''( /6)+12f'(-...
- Let f : [1,∞) → ℝ be a differentiable function defined as f(x)= _1^x f(t)dt+(1-x)( _ex-1)+e. Then the value of f(f(1)) is:
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