Orthocentre (a,b) of triangle (1,2),(2,3),(3,1); I₁=∫_a^b x sin(4x-x²)dx, I₂=∫_a^b sin(4x-x²)dx, find 36 I₁I₂.

∫₀¹ dx√3+x+√1+x = a+b √2+c √3, a,b,c rational, find 2a+3b-4c.

Numerical: Area of {(x,y): x-2y+4≥0, x+2y²≥0, x+4y²≤8, y≥0} = mn, m,n coprime, find m+n.

For 0<a<1, ∫₀^π dx1-2a cosx + a².

∫ x⁸-x²(x¹²+3x⁶+1) tan^-1(x³+1x³) dx.

Numerical: Area of {(x,y):0≤y≤min{2x,6x-x²}} is A, find 12A.

Numerical: f(x)=∫₀^x g(t)log_e(1-t1+t)dt, g continuous odd, ∫_-π/2^π/2 (f(x)+x² cosx/(1+e^x))dx = (πα)²-α, find α.

For x in (-π2, π2), y(x) = ∫ cosec x + sin xcosec x sec x + tan x sin² x dx, lim_x→(π/2)^- y(x)=0, find y(π4).

∫_-π/2^π/2 [x² cos x1+π^x + 1+sin² x1+e^{sin x2023}] dx = (π4)(π+a)-2, find a.

Numerical: Area of part of circle x²+y²=169 below line 5x-y=13 is π α2β-652+(αβ)sin^-1(1213), α,β coprime, find α+β.

∫ sin^(3/2)x+cos^(3/2)x√sin³ x cos³ x sin(x-θ) dx = A √cos θ tan x - sin θ + B √cos θ - sin θ cot x+C, find AB.

Numerical: ∫_π/6^π/3 √1-sin²x dx = α+β √2+γ √3, rational α,β,γ, find 3α+4β-γ.

Numerical: Slope of line 45x+5y+3=0 is 27r1+9r22, find lim_x→3 ∫₃^x 8t² dt(3r22)x² - r2x² - r1x³ -3x.

lim_n→∞ Σ_k=1ⁿ n³(n²+k²)(n²+3k²).

Area bounded by parabola y²=4(x-2) and line y=2x-8.

f:[-π2,π2]→R differentiable, f(0)=12, lim_x→0 x ∫₀^x f(t)dte^x2-1=α, find 8α².

Numerical: 9 ∫₀⁹ [√10x/(x+1)] dx, [.] greatest integer.

y=f(x) thrice differentiable on (-5,5), tangents at (1,f(1)) and (3,f(3)) make angles π6 and π4 with positive x-axis, 27 ∫₁³ ((f'(t))²+1)f''(t)dt = α+β √3, α,β integers, find α+β.

f(x)=x²+3x+a for x≤1, bx+2 for x>1, differentiable on R, find ∫_-2² f(x)dx.

f(x)=x1+x^41/4, g=f o f o f o f, find 18 ∫₀^√2√5 x² g(x) dx.

Numerical: Area enclosed by parabola (y-2)²=x-1, line x-2y+4=0, and positive coordinate axes.

Area of {(x,y): y²≤4x, x<4, xy(x-1)(x-2)(x-3)(x-4)>0, x not 3}.

Numerical: 525 ∫₀^π/2 sin²x cos^11/2x (1+cos^5/2x)^1/2 dx = n √2-64, find n.

Numerical: f(x)=∫_-1^x (e^t-1)¹¹ (2t-1)⁵ (t-2)⁷ (t-3)¹² (2t-10)⁶¹ dt, p=sum of squares of local maxima x, q=sum of local minima x, find p²+2q.

Numerical: f(x)=4^x4^x+2, a not 12, M=∫_f(a)^f(1-a) x sin⁴(x(1-x))dx, N=∫_f(a)^f(1-a) sin⁴(x(1-x))dx, α M=β N, α,β in N, least α²+β².

f(x)=∫_-x^x (|t|-t²)e^-t2dt, g(x)=∫₀^x2 t^1/2e^-tdt, find f(√log_e9)+g(√log_e9).

Area enclosed by y=4x-x² and 3y=(x-4)².

Numerical: |(120π³) ∫₀^π x² sinx cosx/(sin⁴x+cos⁴x) dx|.

∫₀^π/4 x dxsin⁴(2x)+cos⁴(2x).

Area enclosed by xy+4y=16 and x+y=6.

Numerical: ∫_-π/2^π/2 8√2 cos x(1+e^{sin x})(1+sin⁴ x) dx = α π+β log_e(3+2√2), α,β integers, find α²+β².

∫₀¹ (2x³-3x²-x+1)^1/3 dx.

∫₀^π/3 cos⁴x dx = a π+b √3, a,b rational, find 9a+8b.

Numerical: F(x)=∫₀^x t f(t)dt, F(x²)=x⁴+x⁵, find Σ_r=1¹² f(r²).

Numerical: O(0,0), P(a,a²), Q(-b,b²), a>0,b>0 on y=x², S1 area bounded by line PQ and parabola, S2 area of triangle OPQ, min S1S2 = mn, gcd=1, find m+n.

Numerical: Sum of squares of all possible k for which area bounded by parabolas 2y²=kx and ky²=2(y-x) is maximum.

Curves y=1+3x-2x² and y=1x, point (12,2) is an intersection, area = (124)(l √5+m)-n log_e(1+√5), l,m,n in N, find l+m+n.

Numerical: ∫₀^π/4 sin²x/(1+sinx cosx) dx = (1a)log_e(a3)+πb √3, a,b in N, find a+b.

f(x)=∫₀^x (t+sin(1-e^t))dt, find lim_x→0 f(x)x³.

Area of {(x,y): y²≤2x, y≥4x-1}.

∫_-1¹ cos(α x)1+3^x dx = 2π, find a value of α.

∫₀^π/4 136 sin x3 sin x+5 cos x dx.

∫_-π^π 2y(1+sin y)1+cos² y dy.

Numerical: Area enclosed by y=x²-5x and y=7x-x².

Area enclosed between y=x|x| and y=x-|x|.

Beta(m,n)=∫₀¹ x^m-1(1-x)^n-1dx, ∫₀¹ (1-x¹⁰)²⁰dx = a x Beta(b,c), find 100(a+b+c).

Area A enclosed by y=3x, 2y=27-3x, y=3x-x √x, find 10A.

Numerical: r_k=∫₀¹ (1-x⁷)^k dx / ∫₀¹ (1-x⁷)^k+1dx, k in N, find Σ_k=1¹⁰ 17(r_k-1).

I_n = ∫₀¹ (1-x^k)ⁿ dx, n in N, 147 I₂0 = 148 I₂1, find k.

I(x) = ∫ 6sin² x (1-cot x)² dx, I(0)=3, find I(π12).

Parabola y²=4x divides circle x²+y²=5 into two parts, smaller area.

Numerical: Area enclosed by y=min{sinx,cosx} and x-axis between x=-π and x=π be A, find A².

∫ 2-tan x3+tan x dx = (12)(α x + log_e|β sin x + γ cos x|)+C, find α+γβ.

Area of region enclosed by ellipse x²+3y²=18 in first quadrant below line y=x.

limx→π/2 ∫x3(π/2)3 (sin(2t1/3)+cos(t1/3))dtx-π22.

∫_1/4^3/4 cos(2 cot^-1 √(1-x)/(1+x)) dx.

∫_-1² log_e(x+√x²+1) dx.

Let f(x) = 7 tan⁸ x + 7 tan⁶ x - 3 tan⁴ x - 3 tan² x, I₁ = ∫₀^π/4 f(x) dx and I₂ = ∫₀^π/4 x f(x) dx. Then 7 I₁ + 12 I₂ is equal to:

Let f:R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y in R, f'(0) = 4a, f''(x) - 3a f'(x) - f(x) = 0, a > 0. Then the area of the region R = {(x, y) | 0 ≤ y ≤ f(a x), 0 ≤ x ≤ 2} is:

The area of the region inside the circle (x - 2 √3)² + y² = 12 and outside the parabola y² = 2 √3 x is:

Let f(x) = ∫₀^x2 t² - 8t + 15e^t dt, x in R. Then the numbers of local maximum and local minimum points of f, respectively, are:

If ∫ e^x (x - sin^-1 x√1 - x² + sin^-1 x1 - x^23/2 + x1 - x²) dx = g(x) + C, then g(12) equals:

The area of the region enclosed by the curves y = x² - 4x + 4 and y² = 16 - 8x is:

The value of ∫e2e4 (1x) [ e((loge x)2 + 1)-1 / ( e((loge x)2 + 1)-1 + e((6 - loge x)2 + 1)-1 ) ] dx is:

If ∫ 3x + 4x² (x - 2) dx = Ax + B log|x| + C log|x - 2| + ... [as printed, with a, b, c in a set whose symbol is blank/not rendered on the card], then 3(b + c) equals:

If the area of the larger portion bounded between the curves x² + y² = 25 and y = |x - 1| is 14 (b π + c), b, c in [set symbol blank/not rendered on card], then b + c equals ............

If the area of the region {(x, y): -1 ≤ x ≤ 1, 0 ≤ y ≤ a + e^|x| - e^-x, a > 0} is e² + 8e + 1e, then the value of a is:

Let ∫ x³ sin x dx = g(x) + C. If 8(g(π2) + g'(π2)) = α π³ + β π² + γ, α, β, γ in Z, then α + β - γ equals:

If I = ∫₀^π/2 sin^3/2 x / (sin^3/2 x + cos^3/2 x) dx, then ∫₀^2I x sin x cos x / (sin⁴ x + cos⁴ x) dx equals:

The area of the region {(x, y): x² + 4x + 2 ≤ y ≤ |x + 2|} is equal to:

Let f be a differentiable function such that 2(x + 2)² f(x) - 3(x + 2)² = 10 ∫₀^x (t + 2) f(t) dt, x ≥ 0. Then f(2) is equal to ____.

The area of the region enclosed by the curves y = e^x, y = |e^x - 1| and the y-axis is:

If ∫ 2x² + 5x + 9√x² + x + 1 dx = x √x² + x + 1 + α √x² + x + 1 + β log_e |x + 12 + √x² + x + 1| + C, then α + 2 β is equal to ____

If I(m, n) = ∫₀¹ x^m-1 (1 - x)^n-1 dx, m, n > 0, then I(9, 14) + I(10, 13) is:

Let, for some function y = f(x), ∫₀^x t f(t) dt = x² f(x), x > 0 and f(2) = 3. Then f(6) is equal to:

The area, in sq. units, of the region {(x, y): 0 ≤ y ≤ 2|x| + 1, 0 ≤ y ≤ x² + 1, |x| ≤ 3} is:

If ∫_-π/2^π/2 96 x² cos² x1 + e^x dx = π (α π² + β), α, β in [set symbol blank/not rendered on card], then (α + β)² equals:

Let f be a real-valued continuous function defined on the positive real axis such that g(x) = ∫₀^x t f(t) dt. If g(x³) = x⁶ + x⁷, then the value of sum from r = 1 to 15 f(r³) is:

Let f:R → R be a twice differentiable function such that f(2) = 1. If F(x) = x f(x) for all x in R, ∫₀² x F'(x) dx = 6 and ∫₀² x² F''(x) dx = 40, then F'(2) + ∫₀² F(x) dx is equal to:

The area of the region bounded by the curves x(1 + y²) = 1 and y² = 2x is:

If f(x) = ∫ 1x^1/4(1 + x^1/4) dx, f(0) = -6, then f(1) is equal to:

Let the area of the region {(x, y): 2y ≤ x² + 3, y + |x| ≤ 3, y ≥ |x - 1|} be A. Then 6A is equal to:

The ∫ 80 ∫₀^π/4 sin θ + cos θ9 + 16 sin 2θ d θ is equal to:

Let f:(0, ∞) → R be a twice differentiable function. If, for some a ≠ 0, ∫₀¹ f(λ x) d λ = a f(x), f(1) = 1 and f(16) = 18, then 16 - f'(116) is equal to ______.

Let f(x) = ∫₀^x t(t² - 9t + 20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α + β) equals:

Let the area enclosed between the curves |y| = 1 - x² and x² + y² = 1 be α. If 9 α = β π + γ; β, γ are integers, then the value of |β - γ| equals:

If 24 ∫₀^π/4 ( sin|4x - π12| + [2 sin x] ) dx = 2 π + α, where [.] denotes the greatest integer function, then α is equal to ____.

If the area of the region {(x, y): |4 - x²| ≤ y ≤ x², y ≤ 4, x ≥ 0} is (80 √2 / α - β), α, β in N, then α + β = ____.

Let [.] denote the greatest integer function. If ∫₀^e3 [1e^x - 1] dx = α - log_e 2, then α³ is equal to _____.

Let (a, b) be the point of intersection of the curve x² = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the ∫ I = ∫_a^b 9x²1 + 5^x dx is equal to:

Let f:[1, ∞) → [2, ∞) be a differentiable function. If 10 ∫₁^x f(t) dt = 5x f(x) - x⁵ - 9 for all x ≥ 1, then the value of f(3) is:

4 ∫₀¹ 1√3 + x² + √1 + x² dx - 3 log_e (√3) is equal to:

Let f(x) = ∫ x³ √3 - x² dx. If 5 f(√2) = -4, then f(1) is equal to:

The area of the region bounded by the curve y = max{|x|, x|x - 2|}, the x-axis and the lines x = -2 and x = 4 is equal to ____.

The ∫ ∫₀^π 8x dx / (4 cos² x + sin² x) is equal to:

The value of ∫_-1¹ [ (1 + √|x| - x) e^x + √|x| - x e^-x ] / (e^x + e^-x) dx is equal to:

If the area of the region {(x, y): |x - 5| ≤ y ≤ 4 √x} is A, then 3A is equal to ____.

A line passing through the point A(-2, 0) touches the parabola P: y² = x - 2 at the point B in the first quadrant. The area of the region bounded by the line AB, the parabola P and the x-axis is:

Let f(x) + 2 f(1x) = x² + 5 and 2 g(x) - 3 g(12) = x, x > 0. If α = ∫₁² f(x) dx and β = ∫₁² g(x) dx, then the value of 9 α + β is:

If the area of the region bounded by the curves y = 4 - x²4 and y = x - 42 is α, then 6 α equals:

The ∫ ∫₀^π (x + 3) sin x / (1 + 3 cos² x) dx is equal to:

If the area of the region {(x, y): 1 + x² ≤ y ≤ min{x + 7, 11 - 3x}} is A, then 3A is equal to:

If ∫ (1x + 1x³)(cube root of (3x^-24 + x^-26)) dx = -α3(α + 1) (3x^β + x⁷)^{(α + 1)/γ} + C, x > 0, then α + β + γ is equal to ____.

The ∫ ∫_-1^3/2 |π2 - x sin(π x)| dx is equal to:

Let f(x) be a positive function and I₁ = ∫_-1/2^1/2 2x f(2x(1 - 2x)) dx and I₂ = ∫_-1² f(x(1 - x)) dx. Then the value of I₂I₁ is equal to ____.

Area inside ellipse x²+4y²=4 and outside the region bounded by y=|x|-1 and y=1-|x| is ?

Value of ∫_-π/6^π/6 π+4x¹¹1-sin(|x|+π6) dx is ?

6 ∫₀^π |sin 3x+sin 2x+sin x| dx = ?

If area of {(x,y): 1-2x≤y≤4-x², x≥0, y≥0} is αβ, α,β in N, gcd=1, then α+β = ?

Let [.] be greatest integer and f(x)=lim_n→∞ (1n³) Σ_k=1ⁿ [k²3^x]. Then 12 Σ_j=1^∞ f(j) = ?

If ∫₀¹ 4 cot^-1(1-2x+4x²) dx = a tan^-1(2)-b log_e 5, a,b in N, then 2a+b = ?

Line x=-1 divides area of {(x,y): 1+x²≤y≤3-x} in ratio m:n, gcd=1. Then m+n = ?

Let f:[1,∞)→R be differentiable. If 6 ∫₁^x f(t) dt = 3x f(x)-x³-4 for all x≥1, then f(2)-f(3) is ?

If ∫ (sin x)^(-11/2)(cos x)^(-5/2) dx = -(p1q1)(cot x)^(9/2) - (p2q2)(cot x)^(5/2) - (p3q3)(cot x)^(1/2) + (p4q4)(cot x)^(-3/2) + C, p_i,q_i positive integers, gcd=1, then 15 p1 p2 p3 p4q1 q2 q3 q4 = ?

Area of A={(x,y): 4x²+y²≤8 and y²≤4x} is ?

Let [.] be greatest integer. If α=∫₀⁶⁴ (x^(1/3)-[x^(1/3)]) dx, then (1π) ∫₀^απ (sin² θsin⁶ θ+cos⁶ θ) d θ = ?

Let f(x)=∫ (2-x²)e^x√1+x(1-x)^(3/2) dx. If f(0)=0, then f(12) = ?

Value of ∫_π/24^5π/24 dx1+cuberoot(tan 2x) is ?

Let area bounded by y=max{sin x, cos x}, x=0, x=3π2 and x-axis be A. Then A+A² = ?

Let f be twice differentiable, non-negative, with (f(x))²=25+∫₀^x ((f(t))²+(f'(t))²) dt. Mean of f(log_e 1), f(log_e 2), ..., f(log_e 625) is ?

Let I(x)=∫ 3 dx(4x+6)√4x²+8x+3 and I(0)=√34+20. If I(12)=a √2b + c, a,b,c in N, gcd(a,b)=1, then a+b+c = ?

Number of elements in S={x in [0,100]: ∫₀^x t² sin(x-t) dt = x²} is ?

Let f(t)=∫ (1-sin(log_e t)1-cos(log_e t)) dt, t>1. If f(e^(π/2))=e^(π/2) and f(e^(π/4))=α e^(π/4), then α = ?

Let A₁ be bounded area enclosed by y=x²+2, x+y=8 and y-axis in first quadrant; A₂ be bounded area enclosed by y=x²+2, y²=x, x=2 and y-axis in first quadrant. Then A₁-A₂ = ?

If f(x)=e^x+∫₀¹ (y+x e^x) f(y) dy, then e+f(0) = ?

If ∫ (1-5 cos² xsin⁵ x cos² x) dx = f(x)+C, then f(π6)-f(π4) = ?

Area of R={(x,y): xy≤8, 1≤y≤x², x≥0} is ?

Let [.] be greatest integer. Value of ∫_-π/2^π/2 123+[x]3+[sin x]+[cos x] dx is ?

Let f(x)=∫ dxx^(2/3)+2 x^(1/2). If f(0)=-26+24 log_e 2 and f(1)=a+b log_e 3, then a+b = ?

Let f be a polynomial with f(x²+1)=x⁴+5x²+2 for all real x. Then ∫₀³ f(x) dx is ?

Value of Σ_r=1²⁰ √π ∫₀^r x|sin π x| dx is ?

Let f be differentiable with f(x)=1-2x+∫₀^x e^x-t f(t) dt, and g(x)=∫₀^x (f(t)+2)¹⁵ (t-4)⁶ (t+12)¹⁷ dt. If p is sum of local minima and q sum of local maxima of g, then |p+q| = ?

Let [.] be greatest integer. Value of ∫₀³ e^x+e^-xx! dx is ?

If α=∫₀² √3 log₂(x²+4) dx + ∫₂⁴ √2^x-4 dx, then α² = ?

Value of ∫₀^20π (sin⁴ x+cos⁴ x) dx is ?

Let f(x)=∫ 16x+24x²+2x-15 dx. If f(4)=14 log_e 3 and f(7)=log_e(2^α x 3^β), α,β in N, then α+β = ?

If area bounded by 16x²-9y²=144 and 8x-3y=24 is A, then 3(A+6 log_e 3) = ?

Let [.] be greatest integer. If ∫_-2² (|sin x|+[x sin x]) dx = 2(3-cos 2)+β, then β sin(β2) = ?

Area bounded by curves x+3y²=0 and x+4y²=1 is ?

∫₀¹ cot^-1(1+x+x²) dx is ?

Area of R={(x,y): xy≤27, 1≤y≤x²} is ?

Value of ∫₀^∞ log_e xx²+4 dx is ?

Value of ∫_π/6^π/3 (4-cosec² x)/cos⁴ x dx is ?

Let (2^1-a+2^1+a), f(a), (3^a+3^-a) be in AP, and α be the minimum value of f(a). Then ∫_log(α-1)^log(α) dxe^2x-e^-2x is ?

Let f:R→R satisfy f(x)+3f(π2-x)=sin x. Let maximum value of f on R be α. If area bounded by g(x)=x² and h(x)=β x³, β>0, is α², then 30 β³ = ?

Let f:[1,∞)→R be differentiable, defined as f(x)=∫₁^x f(t) dt + (1-x)(log_e x-1)+e. Then f(f(1)) is ?

Value of ∫_-π/4^π/4 32 cos⁴ x1+e^{sin x} dx is ?

Area of {(x,y): 0≤y≤6-x, y²≥4x-3, x≥0} is ?

Area of {(x,y): x²-8x≤y≤-x} is ?

Value of ∫_-1¹ x³+|x|+1x²+2|x|+1 dx is ?

Value of ∫₀² √xx²+x+1(√x+1)√x⁴+x²+1 dx is ?

Let f:(1,∞)→R, f(x)=x-1x+1, fⁱ⁺¹=f(fⁱ), i=1,...,25, f¹=f. If g+f²⁶=0 on (1,∞), then area bounded by y=g(x), 2y=2x-3, y=0, x=4 is ?

If ∫_π/6^π/4 (cot(x-π3)cot(x+π3)+1) dx = α log_e(√3-1), then 9 α² = ?