Match List I with List II:List I (Integrals)List II (Values)A. _0^π/2 2x/^4 x + ^4 x dx I. π/4 B. _0^π/2 x/x + x dx II. π C. _0^2π 1/1 + e^x dx III. π/12 D. _0^π/2 1/9^2 x + 4^2 x dx IV. π/2 Choose the correct answer from the options given below:

A: I = _0^π/2 2x x/^4 x + ^4 x dx . Divide Nr and Dr by ^4 x : I = _0^π/2 2x ^2 x/1 + ^4 x dx . Let t = ^2 x dt = 2x ^2 x dx . I = _0^∞ dt/1 + t^2 = [^-1 t]_0^∞ = π/2 . (A - IV)B: Standard property _0^a f(x) dx = _0^a f(a-x) dx . I = _0^π/2 x/x + x dx = _0^π/2 x/x + x dx 2I = _0^π/2 1 dx = π/2 I = π/4 . (B - I)C: I = _0^2π 1/1 + e^x dx . Use x → 2π - x : I = _0^2π 1/1 + e^-x dx = _0^2π ^xe^x + 1 dx 2I = _0^2π 1 dx = 2π I = π . (C - II)D: I = _0^π/2 ^2 x/9^2 x + 4 dx . Let t = x : I = _0^∞ dt/9t^

Match List I with List II:List I (Integrals)List II (Values)A. _0^π/2 2x/^4 x + ^4 x dx I. π/4 B. _0^π/2 x/x + x dx II. π C. _0^2π 1/1 + e^x dx III. π - Mathematics MCQ Mathematics Mock Test - 11 2026 | ExamTest.live

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Match List I with List II:List I (Integrals)List II (Values)A. \int_{0}^{π /2} \frac{s i n 2 x}{s i n ^{4} x + c o s ^{4} x} d x I. π /4 B. \int_{0}^{π /2} \frac{s i n x}{s i n x + c o s x} d x II. π C. \int_{0}^{2 π} \frac{1}{1 + e ^{s i n x}} d x III. π /12 D. \int_{0}^{π /2} \frac{1}{9 s i n ^{2} x + 4 c o s ^{2} x} d x IV. π /2 Choose the correct answer from the options given below:

A $A - IV, B - II, C - I, D - III$
B $A - II, B - I, C - III, D - IV$
C $A - IV, B - I, C - II, D - III$
D $A - II, B - I, C - IV, D - III$

Solution & Step-by-step Explanation

A: I = \int_{0}^{π /2} \frac{2 s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x . Divide Nr and Dr by cos^{4} x : I = \int_{0}^{π /2} \frac{2 t a n x s e c ^{2} x}{1 + t a n ^{4} x} d x . Let t = tan^{2} x ⟹ d t = 2 tan x sec^{2} x d x . I = \int_{0}^{\infty} \frac{d t}{1 + t ^{2}} = [tan^{- 1} t]_{0}^{\infty} = π /2 . (A - IV)B: Standard property \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x . I = \int_{0}^{π /2} \frac{s i n x}{s i n x + c o s x} d x = \int_{0}^{π /2} \frac{c o s x}{c o s x + s i n x} d x 2 I = \int_{0}^{π /2} 1 d x = π /2 ⟹ I = π /4 . (B - I)C: I = \int_{0}^{2 π} \frac{1}{1 + e ^{s i n x}} d x . Use x \to 2 π - x : I = \int_{0}^{2 π} \frac{1}{1 + e ^{- s i n x}} d x = \int_{0}^{2 π} \frac{e ^{s i n x}}{e ^{s i n x} + 1} d x 2 I = \int_{0}^{2 π} 1 d x = 2 π ⟹ I = π . (C - II)D: I = \int_{0}^{π /2} \frac{s e c ^{2} x}{9 t a n ^{2} x + 4} d x . Let t = tan x : I = \int_{0}^{\infty} \frac{d t}{9 t ^{2} + 4} = \frac{1}{9} \int_{0}^{\infty} \frac{d t}{t ^{2} + ( 2/3 ) ^{2}} = \frac{1}{9} \cdot \frac{3}{2} [tan^{- 1} (3 t /2)]_{0}^{\infty} = \frac{1}{6} \cdot \frac{π}{2} = π /12 . (D - III)Correct Match: A-IV, B-I, C-II, D-III.

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Match List I with List II:List I (Integrals)List II (Values)A. \int_{0}^{π /2} \frac{s i n 2 x}{s i n ^{4} x + c o s ^{4} x} d x I. π /4 B. \int_{0}^{π /2} \frac{s i n x}{s i n x + c o s x} d x II. π C. \int_{0}^{2 π} \frac{1}{1 + e ^{s i n x}} d x III. π /12 D. \int_{0}^{π /2} \frac{1}{9 s i n ^{2} x + 4 c o s ^{2} x} d x IV. π /2 Choose the correct answer from the options given below:

A - IV, B - II, C - I, D - III

A - II, B - I, C - III, D - IV

A - IV, B - I, C - II, D - III

A - II, B - I, C - IV, D - III

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Match List I with List II:List I (Integrals)List II (Values)A. ∫0π/2​sin4x+cos4xsin2x​dx I. π/4 B. ∫0π/2​sinx+cosxsinx​dx II. π C. ∫02π​1+esinx1​dx III. π/12 D. ∫0π/2​9sin2x+4cos2x1​dx IV. π/2 Choose the correct answer from the options given below:

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