Title INTEGRALS Q-1.pdf ✅

CLASS : XIIth SUBJECT : MATHS DATE : DPP NO. : 1 1. Let f(x) = ∫ x 0 |x ― 2|dx, x ≥ 0. Then, f′(x) is a) Continuous and non differentiable at x = 2 b)Discontinuous at x = 4 c) Neither continuous nor differentiable at x = 2 d)Non-differentiable at x = 4 2. If f(t) is an odd function, then ∫ x 0 f(t)dt is a)An odd function b)An even function c) Neither even nor odd d)0 3. ∫ sec x cosec x 2 cot x ― sec x cosec x dx is equal to a)log | sec x + tan x| + c b)log | sec x + cosec x| + c c) 1 2 log | sec 2x + tan 2x| + c d)log | sec 2x + cosec 2x| + c 4. ∫ π 0 θ sin θ 1 + cos2 θ dθis equal to a) π 2 2 b)π 3 3 c) π 2 d)π 2 4 5. The value of [∫ sin2 θ 0 sin―1 φ dφ + ∫ cos2 θ 0 cos―1 φ dφ] is equal to a) π b)π 2 c) π 3 d)π 4 6. ∫ π/2 0 dx 1 + tan3 x is equal to a) π b) π 2 c) π 4 d) 3π 2 7. Let I = ∫ 1 0 sin x x dxand J = ∫ 1 0 cos x x dx. Then, which one of the following is true? a)I > 2 3 and J < 2 b)I > 2 3 and J > 2 c) I < 2 3 and J < 2 d)I < 2 3 and J > 2 8. If f(x) = { |x|, |x ― 2|, ―1 ≤ x ≤ 1 1 < x ≤ 3′ then ∫ 3 ―1 f(x)dx is equal to a) 0 b)1 c) 2 d)4 Topic :-INTEGRALS
9. ∫ 1 x (xn + 1) dx is equal to a) 1 n log ( xn xn + 1 ) +C b) 1 n log ( xn + 1 xn ) c) log ( xn xn + 1 ) +C d)None of these 10. ∫ 1/2 0 |sin π x|dx is equal to a) 0 b)π c) ―π d)1/π 11. The value of the integral ∫ π 0 1 a 2 ― 2a cos x + 1 dx(a > 1), is a) π 1 ― a 2 b) π a 2 ― 1 c) 2π a 2 ― 1 d) 2π 1 ― a 2 12. If I = ∫ 1 0 1 + x 3dx then a)I > 2 b)I ≠ 5 2 c) I > 7 2 d)None of these 13. Assuming that f is everywhere continuous, 1 c ∫ bc ac f( x c )dx is equal to a) 1 c ∫ b a f(x)dx b)∫ b a f(x)dx c) c∫ b a f(x)dx d)∫ bc 2 ac 2 f(x)dx 14. The value of the integral ∫ e x ( 1 ― x 1 + x ) 2 dx is a) e x ( 1 ― x 1 + x 2 ) +c b)e x ( 1 + x 1 + x 2 ) +c c) e x 1 + x 2 +c d)e x (1 ― x) +c 15. Let d dx(F(x)) = e sin x x , x > 0. If ∫ 4 1 3 x e sin x 3 dx = F(k) ―f(1), then one of the possible values of k, is a) 64 b)15 c) 16 d)63 16. The values of ′a′ for which ∫ a 0 (3x 2 + 4x ― 5)dx < a 3 ―2 are a) 1 2 < a < 2 b) 1 2 ≤ a ≤ 2 c) a ≤ 1 2 d)a ≥ 2 17. The value of the integral ∫ log(x + 1) ― log x x(x + 1) dx is a) 1 2 [log(x + 1)] 2 + 1 2 (log x) 2 + log(x + 1)log x +C b) ― [{log(x + 1)} 2 ― (log x) 2 ] + log(x + 1) ∙ log x +C c) 1 2 [log(1 + 1/x)] 2 +C d)None of these 18. The value of ∫ 2π 0 [2 sin x] dx, where [ ∙ ] represents the greatest integral functions, is a) ― 5π 3 b) ―π c) 5π 3 d) ―2π 19. ∫ 1 0 d dx [sin―1 ( 2x 1 + x 2 )] dx is equal to a) 0 b)π c) π 2 d) π 4
20. Let f(x) be a function satisfyingf′(x) = f(x) with f(0) = 1 and g(x) be a function that satisfiesf(x) + g(x) = x 2 . Then, the value of the integral∫ 1 0 f(x)g(x)dx, is a)e ― e 2 2 ― 5 2 b)e + e 2 2 ― 3 2 c) e ― e 2 2 ― 3 2 d)e + e 2 2 + 5 2