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If ∫[(5 tan x) / (tan x – 2)] dx = x + a ln |sin x – 2 cos x| + k, then a =

1) -1 2) -2 3) 1 4) 2 Solution: (4) 2... View Article

∫{sec² x / [(sec x + tan x)^9/2]} dx = (for some arbitrary constant c)

1) 2) 3) 4) Solution: (3)... View Article

∫{(x² – 1) / [x³ √2x⁴ – 2x² + 1]} dx =

1) (- 1 / 2) √[2 - (2 / x2) + (1 / x4)] + c 2) (1 / 2) log (2x4 - 2x2 + 1) + c 3) (1 / 2) √[2 - (2 / x2) + (1 / x4)] + c 4) None of these... View Article

If ∫f (x) dx = Ѱ (x), then ∫x⁵ f (x³) dx =

1) (1 / 3) [x3 Ѱ (x3) - ∫x2 Ѱ (x3) dx] + c 2) (1 / 3) x3 Ѱ (x3) - 3 ∫x2 Ѱ (x3) dx + c 3) (1 / 3) x3 Ѱ (x3) - ∫x2 Ѱ (x3) dx + c 4) (1 / 3) [x3... View Article

∫{1 / [x³ (x³ – 1)]} dx =

1) (1 / 3) log |(x3) / (x3 - 1)| + c 2) (1 / 3) log |(1 - x3) / (x3)| + c 3) log |(x3) / (x3 - 1)| + c 4) (1 / 3) log |x - 1| + (1 / 2x2) - (1... View Article

∫sec³ x dx =

∫[sec² (sin^-1 x) / (√1 – x²)] dx =

1) sin (tan-1 x) + c 2) tan (sec-1 x) + c 3) tan (sin-1 x) + c 4) - tan (cos-1 x) + c Solution: (3) tan (sin-1 x) + c ∫[sec2 (sin-1 x) / (√1... View Article

∫[(x dx) / (1 + x)^3/2] =

1) 2 [(2 + x) / (√1 + x)] + c 2) [(2 + x) / (√1 + x)] + c 3) (3 / 2) [x / √1 - x] + c 4) (3 / 2) [(2 + x) / (√1 + x)] + c Solution: (1) 2 [(2... View Article

For any integer n ≥ 2, let l_n = ∫tanⁿ x dx. If l_n = (1 / a) tan^n-1 x – bl_n-2 for n ≥ 2, then the ordered pair (a, b) =

1) [(n - 1), (n - 1) / (n - 2)] 2) [(n - 1), (n - 2) / (n - 1)] 3) (n, 1) 4) (n - 1, 1) Solution: (4) (n - 1, 1)... View Article

If ∫dx / [√(sin³ x cos x)] = g (x) + c, then g (x) =

1) - 2 / (√cot x) 2) - 2 / √tan x 3) 2 / √cot x 4) 2 / √tan x Solution: (2) - 2 / √tan x... View Article

∫{(√(5 + x²)) / (x⁴)} dx =

1) (1 / 15) [1 + (5 / x2)]3/2 + c 2) (- 1 / 15) [1 + (1 / x2)]3/2 + c 3) (- 1 / 15) [1 + (5 / x2)]3/2 + c 4) (1 / 15) [1 + (1 / x2)]3/2 + c... View Article

∫e^x (sin x + 2 cos x) sin x dx =

1) ex cos x + c 2) ex sin x + c 3) ex sin2 x + c 4) ex sin 2 x + c 5) ex (cos x + sin x) + c Solution: (3) ex sin2 x + c ∫ex (sin x + 2... View Article

∫[xe^x / (1 + x)²] dx =

1) [- ex / (x + 1)] + c 2) [ex / (x + 1)] + c 3) [xex / (x + 1)] + c 4) [- xex / (x + 1)] + c 5) [ex / (x + 1)2] + c Solution: (2) [ex / (x... View Article

∫[(1 + x)^{e^x} / sin² (xe^x)] dx =

1) - cot (ex) + c 2) tan (xex) + c 3) tan (ex) + c 4) cot (xex) + c 5) - cot (xex) + c Solution: (5) - cot (xex) + c... View Article

∫{1 / [x² (x⁴ + 1)^¾]} dx =

1) [- (x4 + 1)¾ / x] + c 2) [- (1 - x4)1/4 / 2x] + c 3) [- (1 + x4)1/4 / x] + c 4) [- (1 + x4)1/4 / x2] + c 5) [- (1 + x4)1/2 / x] + c... View Article

If linear function f (x) and g (x) satisfy ∫[(3x – 1) cos x + (1 – 2x) sin x] dx = f (x) cos x + g (x) sin x + c, then

1) f (x) = 3 (x - 1) 2) f (x) = 3x - 5 3) g (x) = 3 (x - 1) 4) g (x) = 3 + x Solution: (3) g (x) = 3 (x - 1)... View Article

Biodiversity conservation during a global crisis: RSTV- Big Picture

Biodiversity conservation during a global crisis:- Download PDF Here Anchor:- Teena Jha Participants : Vivek Saxena, Country Representative,... View Article

25 May 2021: UPSC Exam Comprehensive News Analysis

CNA 25th May 2021:- Download PDF Here TABLE OF CONTENTS A. GS 1 Related SOCIAL ISSUES 1. Provide food, rations to stranded workers: SC B. GS 2... View Article

24 May 2021: PIB Summary & Analysis

May 24th, 2021, PIB:- Download PDF Here TABLE OF CONTENTS 1. Cyclonic Storm ‘Yaas’ 2. Hallmarking of Gold Jewellery in India 3. Technology... View Article

24 May 2021: The Hindu Analysis

UPSC Current Affairs Preparation: The Hindu Analysis Watch expert analysis of ‘The Hindu’ dated 24th May 2021. Important news and views to boost... View Article

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