Integral Calculus — practice questions

175 curated practice questions from Integral Calculus (Mathematics JEE Advanced, NTA Exams) — 120 hard, 55 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If f: R R is a function satisfying the following : (i) f (– x ) = – f ( x ) (ii) f ( x + 1) = f ( x ) + 1 (iii) f ( 1 x…
• If I = x^ 3 +x x^ 4 -9 ~d x then I equals :
• If ( x 2 + x ) dx = x log |x | + ( x + 1) log | x +1| + k, then k equals
• The area of the region between the curves y= 1+ x x and y= 1- x x bounded by the lines x = 0 and x= 4 is
• _ 0 ^ / 2 f( 2 x) x d x=k _ 0 ^ / 4 f( 2 x) x d x where equals
• x-1 (x+1) x (x^ 2 +x+1 ) d x is
• ( x 3 + 3 x 2 + 3 x + 3 + ( x + 1) cos ( x + 1)) dx is equal to
• If is integrable over [1,2] , then _ 1 ^ 2 f(x) d x is equal to
• If I _ n = ( x)^ n ~d x then I n + n I n–1 =
• Evaluate (x+2) d x (x^ 2 +3 x+3 ) x+1
• If ( 1- x 1+ x )^ 1 / 2 d x x =2 ^ -1 x -f(x)+c then f (x) equals.
• Assertion : 1 _ 0 ^ / 2 x x 2 Reason : If is continuous in and m and are greatest and least value of in , then l(b-a) _…
• If f(x)=e^ g(x) and g(x)= _ 2 ^ x t 1+t^ 4 d t then f^ (2) has the value equal to
• The value of _ ^ 2 [2 x] d x where [⋅] represents the greatest integer function is
• The ratio in which the curve y=x^ 2 divides the region bounded by the curve; y= ( x 2 ) & the x- axis as varies from 0…
• _ h 0 _ a ^ x+h ^ 2 t d t- _ a ^ x ^ 2 t d t h equals to
• We can derive reduction formula for the integration of the form ^ n x d x, ^ n x d x, ^ n x d x and other integrals of…
• _ / 4 ^ 3 / 4 d x 1+ x is equal to
• The value of the definite integral _ 0 ^ 1 (1+ e ^ - x ^ 2 ) d is :
• If F(x)= 1 x^ 2 _ 4 ^ x (4 t^ 2 -2 F^ (t) ) d t then F^ (4) equals –
• Assertion : e^ ^ -1 x (1- x 1-x^ 2 ) d x=e^ ^ -1 x 1-x^ 2 +c Reason : e ^ g(x) (g^ ( x ) f( x )+f^ ( x ) ) dx = e ^ g (…
• If _ 0 ^ x^ 2 (1+ x)^ 2 d x=A then _ 0 ^ 2 x^ 2 ^ 2 (x / 2) (1+ x)^ 2 d x=
• The value of _ ^ 2 [2 x] d x where [.] represents the greatest integral functions, is :
• Let f : R → R and g : R → R be continuous functions. Then the value of integral _ - 2 ^ 2 [f(x)+f(-x)][g(x)-g(-x)] d x
• The value of the function f(x)=1+x+ _ 1 ^ x ( ^ 2 t+2 t ) d t where f^ (x) vanishes is
• _ - ^ 2 x(1+ x) 1+ ^ 2 x d x is
• The value of the integral ^ 3 x+ ^ 5 x ^ 2 x+ ^ 4 x d x is
• If denotes the integeral part of , then _ 0 ^ 1 ( x) ([2 x] ) d x=
• If f(x)= array ll e^ x x, & for |x| 2 2, & otherwise array . then _ -2 ^ 3 f(x) d x
• Let g(x)= _ 0 ^ x f(t) d t , where f is such that 1 2 f(t) 1 for t [0,1] and 0 f(t) 1 2 for t [1,2] then g (2)…
• The area enclosed by the curves y = sin x + cos x and y = |cos x – sin x | over the interval [0, 2 ] is
• (1+x) x (x^ 2 +2 x ) ^ 2 x-(1+x) 2 x d x = 1 2 _ e | t+1 t-1 |+c where t is
• _ n 1 n _ r=1 ^ 2 n r n^ 2 +r^ 2 equals
• Evaluate : (1+x) x (1+x e^ x )^ 2 d x
• If g(x) d x=g(x) then g(x) (f(x)+f^ (x) ) d x is equal to
• The area of the closed figure bounded by the curves y= x , y= 4-3 x y=0 is
• 1 x^ 2 -1 ( x-1 x+1 ) dx equals
• We can derive reduction formula for the integration of the form ^ n x d x, ^ n x d x, ^ n x d x and other integrals of…
• d x 5+4 x = ^ -1 (m x 2 )+c then
• If I_ n = _ 0 ^ 1 d x (1+x^ 2 )^ n , n N, then which of the following statements hold good ?
• Let f : [–1, 2] → [0, ∞ ) be a continuous function such that f ( x ) = f (1 – x ) for all x ∈ [–1, 2]. Let R_ 1 = _ -1…
• Let J= e^ x e^ 4 x +e^ 2 x +1 d x, I= e^ -x e^ -4 x +e^ -2 x +1 d x Then, for an arbitrary constant C , the value of J…
• Let F ( x ) be an indefinite integral of sin 2 x . Statement-1 : The function F ( x ) satisfies F ( x + π) = F ( x )…
• If m= _ -2 ^ 0 | x| [ x ]+ 1 2 d x and n= _ 0 ^ 2 | x| [ x ]+ 1 2 d x where represents greatest integer function, then
• If the integrand is a rational function of x and fractional powers of a linear fractional function of the form a x+b c…
• If I= _ 0 ^ 1 e^ t t+1 d t then _ a-1 ^ a e^ -t t-a-1 d t=
• The area enclosed by y=x^ 3 , its normal at (1,1) and x- axis is equal to
• If I_ n = ^ n x d x then I_ n - ( n-1 n ) I_ n-2 =
• If I_ n = _ 1 ^ e ( _ e x )^ n d x is a positive integer), then I_ 2012 +(2012) I_ 2011 =
• Area of the region bounded by y e^ x and y x , is
• The area of the region enclosed between the curves 7 x^ 2 +9 y+9=0 and 5 x^ 2 +9 y+27=0 is
• If stands for the integral part of , then _ 0 ^ 5 / 12 [ x] d x=
• If a b and a f(x)+b f ( 1 x )= 1 x -5 for all x 0 , then _ 1 ^ 2 f(x) d x= 1 a^ 2 -b^ 2 [a( 2- )+ ( b 2 ) ] where - is…
• Consider the curve defined implicity by the equation y^ 2 -2 y e^ ^ -1 x +x^ 2 -1+[x]+e^ 2 ^ -1 x =0, Where denotes the…
• If f(x)= _ ^ 2 / 16 ^ x^ 2 x t 1+ ^ 2 t d t then f^ ( ) is equal to
• Using integral _ 0 ^ / 2 ( x) d x =- _ 0 ^ / 2 ( x) d x=- 2 2 _ 0 ^ / 2 ( x) d x=0 and _ 0 ^ / 4 (1+ x) d x= 8 2…
• The option(s) with the values of a and L that satisfy the following equation is(are) _ 0 ^ 4 e^ t ( ^ 6 a t+ ^ 4 a t )…
• _ 0 ^ x^ 2 2 x [( / 2) x] 2 x- d x=
• The area bounded by curve y=e x x and y= x e x is –
• The value of _ 0 ^ / 2 d x 1+ ^ 3 x is
• If g(x)= _ 0 ^ x ^ 4 t d t then equals
• If f (x) = A sin ( x 2 ) + B, f^ ( 1 2 )= 2 and _ 0 ^ 1 f( x ) dx = 2 ~A then constants A and B are :
• The area bounded by the curve y = ( x + 1) 2 , y = ( x – 1) 2 and the line is
• Area of the region bounded by the curve y = e x and lines x = 0 and y = e is
• If f ( x ) is a polynomial function of the n th degree, then e^ x f(x) d x is equal to
• If I_ n = _ - ^ n x (1+ ^ x ) x d x, n=0,1,2, . , then
• d x x+ x+ x+ cosec x is equal to
• The maximum value of _ a-1 ^ a+1 e^ -(x-1)^ 2 d x is attained (a is real) at
• The value of the integral _ - / 2 ^ / 2 (x^ 2 + +x -x ) x d x is
• If I= _ k=1 ^ 98 _ k ^ k+1 k+1 x(x+1) d x , then
• The value of the definite integral _ 0 ^ 1 (1+e^ -x^ 2 ) d x is
• Let f : R → R and g : R → R be continuous functions. Then the value of the integral _ - / 2 ^ / 2…
• The function F(x)= _ / 6 ^ x (4 t+3 t) d t attains least value on [π/4, 3π/4] at x equals.
• If primitive of sin (log x) is f (x) (sin (g(x)) – cos (h(x)) + c then
• Let f^ (x)= 192 x^ 3 2+ ^ 4 x for all x ∈ R with f ( 1 2 )=0 . If m _ 1 / 2 ^ 1 f(x) d x M then the possible values of…
• If f(t)= array ll at -1 & t so that _ 0 ^ x f(x) d x is differentiable for all x 0 is
• The value of _ - ^ ^ 2 x 1+a^ x d x , a > 0 is
• For any integer n, the integral _ 0 ^ e^ ^ 2 x ^ 3 (2 n+1) x d x has the value :
• If _ x 0 f(x) x^ 2 exists finitely and _ x 0 (1+x+ f(x) x )^ 1 / x =e^ 3 where f ( x ) = a x 2 +b x +c then f(x) _ e x…
• The following integral is equal to
• x^ 2 -1 (x^ 4 +3 x^ 2 +1 ) ^ -1 ( x^ 2 +1 x ) d x is equal to
• If I = ^ 7 x ~d x then I equals
• The value of the integral _ 0 ^ 1 1-x 1+x d x is
• If (x^ 3 -2 x^ 2 +5 ) e^ 3 x dx = e 3x (Ax 3 + Bx 2 + Cx + 13/9) then which of the following statement is incorrect :
• The value of the integral _ e^ -1 ^ e^ 2 | _ e x x | d x is
• _ 1 / e ^ x t 1+t^ 2 d t+ _ 1 / e ^ x dt t (1+t^ 2 ) =
• Let f be a positive function. Let I_ 1 = _ 1-k ^ k x f[x(1-x)] d x I_ 2 = _ 1-k ^ k f[x(1-x)] d x where 2 k – 1 > 0…
• _ 0 ^ 2 e^ | x| x 1+e^ x d x=
• If the integrand is a rational function of x and fractional powers of a linear fractional function of the form a x+b c…
• x - a x + a dx
• For 0 f(x)= _ n (1+x) (1+x^ 2 ) (1+x^ 4 ) (1+x^ 2^ n ) then f(x) 1-x log e xdx equals
• For each positive integer n, define f_ n (x)= Min ( x^ n n! , (1-x)^ n n! ) for 0 x 1 . Let I_ n = _ 0 ^ 1 f_ n (x) d x…
• The area of the closed figure bounded by the curves y= x ; y=1+ 2 x x= 2 is
• If I= ( 2 ) d then I equals
• _ 3 ^ 29 [3] (x-2)^ 2 3+ [3] (x-2)^ 2 d x=
• The area bounded by y=2-|2-x| and y= 3 |x| is
• If _ 0 ^ x f(t) d t=x+ _ x ^ 1 t f(t) d t then the value of f (1) is
• For any integer n , the integral _ 0 ^ e^ ^ 2 x ^ 3 (2 n+1) x d x has the value
• Consider the function f(x)= _ 0 ^ x [t] d t where x>0 and is the integral part of . Then
• Let = _ 0 ^ 1 d x 1+x^ 3 , p= _ n [ _ r=1 ^ n (n^ 3 +r^ 3 ) n^ 3 n ]^ 1 / n then ln p is equal to
• The area enclosed by the curves y= 4-x^ 2 , y 2 ( x 2 2 ) and x- axis is divided by y- axis in the ratio
• The value of the integral _ 0 ^ / 2 x x + x d x is
• If f(2-x)=f(2+x) and f(4-x)=f(4+x) and is a function for which _ 0 ^ 2 f(x) d x=5 then _ 0 ^ 50 f(x) d x is equal to
• Assertion : _ 0 ^ x x ^ 2 x d x= 2 _ 0 ^ x ^ 2 x d x Reason : _ a ^ b x f(x) d x= a+b 2 _ a ^ b f(x) d x
• Assertion : x^ 2 -1 x^ 2 e^ x^ 2 +1 x d x=e^ x^ 2 +1 x +c Reason : f(x) e^ f(x) d x=e^ f(x) +c
• Statement-I : The value of the integral _ / 6 ^ / 3 d x 1+ x is equal to π/6. Statement-II : _ a ^ b f(x) d x= _ a ^ b…
• ( p x q x) d x is equal to (where p, q ∈ Z)
• If I= ^ 3 2 x ^ 5 x d x and f(x) = (cot x) 3/2 , g(x) = (cot x) 5/2 , then I equals
• The area bounded by the curve y=e^ x and the lines y=| x -1|, x=2 is given by
• The value of _ 2 ^ 3 x x^ 2 x^ 2 + ( 6-x^ 2 ) d x is
• Let f(x)= array ll _ 0 ^ x 5+|1-y| d y & if x>2 5 x+1 & if x 2 array . Then
• Area of the region is equal to
• Let a , b , c be non zero real numbers such that _ 0 ^ 1 (1+ ^ 8 x ) (a x^ 2 +b x+c ) d x= _ 0 ^ 2 (1+ ^ 8 x ) (a x^ 2…
• Consider the curve defined implicity by the equation y^ 2 -2 y e^ ^ -1 x +x^ 2 -1+[x]+e^ 2 ^ -1 x =0, Where denotes the…
• If the line x = α divides the area of region R = ( x , y ) ∈ R 2 : x 3 ≤ y ≤ x , 0 ≤ x ≤ 1 into two equal parts, then
• _ 0 ^ _ 5 5 e^ x e^ x -1 e^ x +3 d x=
• If I = x^ 2 + a ^ 2 x^ 4 - a ^ 2 x^ 2 + a ^ 4 ~d x then I =
• _ 0 ^ 1 / 3 d x (2 x^ 2 +1 ) x^ 2 +1 =
• Let g(x) be an antiderivative of f (x). Then ln (1+(g(x)) 2 ) is an antiderivative for :
• _ -1 / 3 ^ 1 / 3 ^ -1 ( 2 x 1+x^ 2 )+ ^ -1 ( 2 x 1-x^ 2 ) e^ x +1 d x=
• For which of the following values of m , is the area of the region bounded by the curve y = x – x 2 and the line y = mx…
• The area bounded by the curve y=3+2 x-x^ 2 , y=0 & the ordinate at x=1 x=4 is
• Let the functions f : ℝ → ℝ and g : ℝ → ℝ be defined by f ( x ) = e x – 1 – e –| x – 1| and g ( x ) = ( e x – 1 + e 1 –…
• If I_ m, n = _ 0 ^ 1 t^ m (1+t)^ n d t then the expression for I ( m, n ) in terms of I m + 1, n – 1 is
• If _ 0 ^ x e^ 2 x e^ -z^ 2 d z=f(x) _ 0 ^ x e^ -z^ 2 / 4 d z then e^ x ( _ e (f(x))+ x 2 ) d x=
• Consider the equation _ 1 ^ e ( _ e x )^ 1 / 2 x (a- ( _ e x )^ 3 / 2 )^ 2 d x=1, a (- , 0) (1, ) . Which of the…
• Evaluate I= ( a+x a-x + a-x a+x ) d x
• Using integral _ 0 ^ / 2 ( x) d x =- _ 0 ^ / 2 ( x) d x=- 2 2 _ 0 ^ / 2 ( x) d x=0 and _ 0 ^ / 4 (1+ x) d x= 8 2…
• If _ 0 ^ x f(t) d t=x+ _ x ^ 1 t f(t) d t then f(1) is
• The integral _ -1 2 ^ 1 2 [[x]+ ( 1+x 1-x ) ] d x equals
• If f(x)= x x [0, 2 ], f(x)+f( -x)= 2 x ( 2 , ] and f(x)=f(2 -x) x ( , 2 ] , then the area enclosed by y=f(x) and x…
• Consider the curve defined implicity by the equation y^ 2 -2 y e^ ^ -1 x +x^ 2 -1+[x]+e^ 2 ^ -1 x =0, Where denotes the…
• Value of the parameter a such that the area bounded by y=a^ 2 x^ 2 +a x+1 , co-ordinate axes and the line x=1 , attains…
• x^ 2 -1 x^ 3 2 x^ 4 -2 x^ 2 +1 d x=
• We can derive reduction formula for the integration of the form ^ n x d x, ^ n x d x, ^ n x d x and other integrals of…
• Evaluate : x -1 x x +1 dx
• Let f(x)= x (1+x^ n )^ 1 / n for n ≥ 2 and Then x^ n-2 g(x) d x equals
• Let the straight line x = b divide the area enclosed by y = (1 – x ) 2 , y = 0 and x = 0 into two parts R 1 (0 ≤ x ≤ b…
• If is differentiable & defined on R^ + such that _ 0 ^ t^ 2 x f(x) d x= 2 3 t^ 5 then f(4 / 25)=
• Evaluate : 1+x^ 2 n x^ 2 n (1+x^ 2 n )-2 n / n x x^ 2 n+1 d x
• If I= d x (a^ 2 -b^ 2 x^ 2 )^ 3 / 2
• The area bounded by the curves and x -axis in the 1 st quadrant is
• We can derive reduction formula for the integration of the form ^ n x d x, ^ n x d x, ^ n x d x and other integrals of…
• Consider the integrals I_ 1 = _ 0 ^ 1 e^ -x ^ 2 x d x, I_ 2 = _ 0 ^ 1 e^ -x^ 2 ^ 2 x d x I_ 3 = _ 0 ^ 1 e^ - x^ 2 2 ^ 2…
• If I = (2 x +3) dx ( x ^ 2 +2 x +3 ) x ^ 2 +2 x +4 , then I equals
• Let S be the area of the region enclosed by y = e – x 2 , y = 0, x = 0 and x = 1 . Then
• (x^ 2 -1 ) d x 2 x x^ 4 +4 x^ 3 -6 x^ 2 +4 x+1 is
• _ 0 ^ x (1+x) (1+x^ 2 ) d x
• The area of the region ( x, y ) : xy ≤ 8, 1 ≤ y ≤ x 2 is
• Assertion : If 1 f(x) d x=2 log | f (x)|+c, then f(x)= x 2 Reason : When f (x) = x 2 , then 1 f(x) d x= 2 x d x=2 |x|+c
• If f(x)= Lim _ n e x tan (1/n) log (1/n) and f(x) d x [3] ^ 11 x x = g(x) + c then
• If I = e ^ x ( x x+ x) d x then I equals :
• Let f(x)= e^ x 1+e^ x I_ 1 = _ f (- a ) ^ f ( a ) x g(x(1-x)) d x I_ 2 = _ f (- a ) ^ f ( a ) g(x(1-x)) d x then I _ 2…
• If for a real number y , [ y ] is the greatest integer less than or equal to y , then the value of the integral _ / 2 ^…
• If f ( x ) is differentiable and _ 0 ^ t^ 2 x f(x) d x= 2 5 t^ 5 then f ( 4 25 ) equals
• x- x ( x+ x) x x+ ^ 2 x ^ 2 x d x=
• Using integral _ 0 ^ / 2 ( x) d x =- _ 0 ^ / 2 ( x) d x=- 2 2 _ 0 ^ / 2 ( x) d x=0 and _ 0 ^ / 4 (1+ x) d x= 8 2…
• If y= x^ 2 -x+1 and for n > 1, I_ n = x^ n / y d x and aI 3 + bI 2 + cI 1 = x 2 y, then (a, b, c) is equal to
• Let T > 0, be a fixed real number. Suppose f is a continuous function. Such that for all x ∈ R . f ( x + T ) = f ( x )…
• A function which satisfies the relation f(x)=e^ x + _ 0 ^ 1 e^ x f(t) d t then is
• If d x (x^ 2 -4 ) x+1 = 1 k | x+1 - 3 x+1 + 3 | - 1 2 ^ -1 x+1 +c then k equals
• The area enclosed between the curves y = ax 2 and x = ay 2 ( a > 0) is 1 sq. unit, then the value of a is
• _ 0 ^ 1 d x (5+2 x-2 x^ 2 ) (1-e^ 2-4 x ) =
• _ 0 ^ 1 / 2 e^ x ( ^ -1 x- x (1-x^ 2 )^ 3 / 2 ) d x=
• Evaluate : x d x (2 x+A)+ A
• If I_ 1 = _ -4 ^ -5 e^ (x+5)^ 2 d x and I_ 2 =3 _ 1 / 3 ^ 2 / 3 e^ 9[x-(2 / 3)]^ 2 d x then the value of I _ 1 + I _ 2…
• The area of the plane figure bounded in first quadrant by y=x^ 1 / 3 ; y=-x^ 2 +2 x+3 ; y=2 x-1 and the axis of…
• The area bounded by the curves y = f ( x ), the x -axis and the ordinates x = 1, x = b is ( b – 1) sin (3 b + 4). Then…
• The value of _ - / 2 ^ / 2 x^ 2 x 1+e^ x d x is equal to
• The slope of the tangent to a curve y = f ( x ) at [ x , f ( x )] is 2 x + 1. If the curve passes through the point (1…
• Evaluate : x d x 3 ^ 2 x+4 ^ 2 x
• e ^ 2 ^ -1 x (1+ x )^ 2 (1+ x ^ 2 ) dx is equal to
• The area bounded by the curve y=x(1- x) and positive x- axis between x=e^ -1 and x=e is
• If I= _ 0 ^ / 2 e^ - x d x where (0, ), then
• _ 0 ^ ^ 2 x ( ^ -1 t ) d t+ _ 0 ^ ^ 2 x ( ^ -1 t ) d t=

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