Definite Integration questions
424 published questions with worked solutions.
- Let f : ℝ→ℝ be such that f(xy)=f(x)f(y), for all x,y∈ℝ and f(0)≠0. Let g : [1,∞)→ℝ be a differentiable function such that...
- The value of the integral _ -1 ^ 1 ( x^3+|x|+1 x^2+2|x|+1 )dx is equal to:
- The value of the integral _ - /4 ^ /4 ( 32 ^4x 1+e^ x )dx is:
- Let _0^x 1 - (y'(t))^2 dt = _0^xy(t)dt, 0 ≤ x ≤ 3, y ≥ 0, y(0) = 0. Then at x = 2, y″ + y + 1 is equal to :
- Let _ n ( n n^4+1 - 2n (n^2+1) n^4+1 + n n^4+16 - 8n (n^2+4) n^4+16 + + n n^4+n^4 - 2n n^2 (n^2+n^2) n^4+n^4 ) be π/k, using only...
- Let f(x) = _0^xg(t) _e ( 1-t 1+t )dt, where g is a continuous odd function. If _ - /2 ^ /2 (f(x) + x^2 x 1 + e^x )dx = ( )^2 - ,...
- If the value of the integral _ - /2 ^ /2 ( x^2 x 1+ ^x + 1+ ^2x 1+e^ x^ 2003 )dx = 4( + a) - 2, then the value of a is-
- Let f : ℝ → ℝ be a function defined by f(x) = x (1+x^4)^ 1/4 , and g(x) = f(f(f(f(x)))). Then 18 _0^ 2 5 x^2g(x)dx is equal to
- Let f : ℝ → ℝ be defined as f(x) = ae²ˣ + beˣ + cx. If f(0) = −1, f′(log_e 2) = 21 and _0^ _e4 (f(x) - cx)dx = 39 2, then the...
- The value of _ n _ k=1 ^n n^3 (n^2 + k^2)(n^2 + 3k^2) is:
- The value of 9 _0^9 [ 10x x+1 ]dx, where [t] denotes the greatest integer less than or equal to t, is ________.
- Let f, g : (0, ∞) → ℝ be two functions defined by f(x) = _ -x ^x(|t| - t^2)e^ -t^2 dt and g(x) = _0^ x^2 t^ 1/2 e^ -t dt. Then,...
- | 120 ^3 _0^ x^2 x x ^4x + ^4x dx | is equal to ________.
- Let f : ℝ → ℝ be a function defined by f(x) = 4ˣ/(4ˣ + 2) and M = _ f(a) ^ f(1-a) x ^4(x(1-x))dx, N = _ f(a) ^ f(1-a)...
- Let f:(0,2) R be defined as f(x)= _2 (1+ ( x 4 ) ). Then, _ n 2 n (f ( 1 n )+f ( 2 n )+.....+f(1) ) is equal to _______.
- Let f:R R be a continuous function such that f(x) + f(x + 1) = 2, for all x . If I_1= _0^8f(x)dx I_2= _ -1 ^3f(x)dx, then the...
- Let f : R R be defined as f(x) = e^ -x x. If F : [0, 1] R is a differentiable function such that F(x) = _0^xf(t)dt, then the...
- If the integral _0^ 10 [ 2 x] e^ x-[x] dx= e^ -1 + e^ - 1 2 + (given), where , , are integers and [x] denotes the greatest...
- The value of _ n [r]+[2r]+ +[nr] n^2 , where r is non-zero real number and [r] denotes the greatest integer less than or equal to...
- Let g(x) = _0^xf(t)dt, where f is continuous function in [0, 3] such that 1 3 f(t) 1 for all t [0, 1] and 0 f(t) 1 2 for all t...
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