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∫[(1 + x⁴) / (1 – x⁴)^3/2] dx =

1) [x / √1 - x4] + c 2) [- x / √1 - x4] + c 3) [2x / √1 - x4] + c 4) [- 2x / √1 - x4] + c Solution: (1) [x / √1 - x4] + c... View Article

If ∫uv’’ dx = uv’ – vu’ + a, then a =

1) ∫uv’’ dx 2) ∫u’v dx 3) ∫uv’ dx 4) ∫u’’ dx Solution: (1) ∫uv’’ dx ∫uv’’ dx = uv’ - ∫u’v’ dx = uv’ - uv + ∫u’’v dx... View Article

If ∫[(x² – 2) / (x³ √x² – 1))] dx =

1) [(x2) / √x2 - 1)] + c 2) [(- x2) / √x2 - 1)] + c 3) [(√x2 - 1) / (x2)] + c 4) [- (√x2 - 1) / (x2)] + c Solution: (4) [- (√x2 - 1) / (x2)] + c... View Article

If ∫dx / [x⁴ + x³] = [A / x²] + (B / x) + log |x / (x + 1)| + c, then

1) A = 1/2, B = 1 2) A = 1, B = 1/2 3) A = - (1/2) , B = 1 4) A = 1, B = 1 Solution: (3) A = - (1/2) , B = 1... View Article

If ∫[log (log x) + 1 / (log x)²] dx = x [f (x) – g (x)] + c

1) f (x) = log (log x); g (x) = 1 / log x 2) f (x) = log x, g (x) = 1 / log x 3) f (x) = 1 / [x log x]; g (x) = log (log x) 4) f (x) = 1 / [x... View Article

If l_n = ∫sinⁿ x dx, then nl_n – (n – 1) l_n-2 =

1) sinn-1 x cos x 2) cosn-1 x sin x 3) - sinn-1 x cos x 4) - cosn-1 x sin x Solution: (3) - sinn-1 x cos x... View Article

∫[log (x + √1 + x²) / √1 + x²] dx =

1) [log (x + √1 + x2)]2 + c 2) x log (x + √1 + x2) + c 3) (1 / 2) log (x + √1 + x2) + c 4) (1 / 2) [log (x + √1 + x2)]2 + c 5) (x / 2) log (x... View Article

If ∫[(x + 2) / (2x² + 6x + 5)] dx = P ∫[(4x + 6) / (2x² + 6x + 5)] dx + (1 / 2) ∫dx / (2x² + 6x + 5), then the value of P is

1) 1/3 2) 1/2 3) 1/4 4) 2 5) 1 Solution: (3) ¼... View Article

∫[dx / (√1 – e^2x)] =

∫[(cos x + x sin x) / (x² + x cos x)] dx =

∫[(1 + x) / (x + e^-x)] dx =

1) log |(x - e-x)| + c 2) log |(x + e-x)| + c 3) log |(1 + xex)| + c 4) (1 + xe-x)2 + c 5) log |(1 - xex)| + c Solution: (3) log |(1 + xex)|... View Article

∫[(x² + 1) √(x + 1)] dx =

1) 2) 3) 4) 5) Solution: (2)... View Article

∫e^x [(2 / x) – (2 / x²) dx] =

1) (ex / x) + c 2) (ex / 2x2) + c 3) (2ex / x) + c 4) (2ex / x2) + c Solution: (3) (2ex / x) + c ∫ex [(2 / x) - (2 / x2) dx] = 2 ∫[ex (x -... View Article

∫{log √x / (3x)} dx =

1) (1 / 3) (log √x)2 + c 2) (2 / 3) (log √x)2 + c 3) (2 / 3) (log x)2 + c 4) (1 / 3) (log x)2 + c Solution: (1) (1 / 3) (log √x)2 + c ∫{log... View Article

25 May 2021: PIB Summary & Analysis

May 25th, 2021, PIB:- Download PDF Here TABLE OF CONTENTS 1. Vesak or Buddha Purnima 2. National Mission on use of Biomass in Thermal Power... View Article

Mayflower 400 – World’s First Artificial Intelligence Ship

The Mayflower 400 is the world’s first artificial intelligence managed vessel to sail on a transatlantic voyage. It will track aquatic mammals,... View Article

Global Report on Food Crises (GRFC 2021)

The fifth edition of the Global Report on Food Crises (GRFC 2021) was released on May 5, 2021. A high-level virtual event was held by the Global... View Article

Cairn Tax Dispute

In December 2020, the Permanent Court of Arbitration (PCA) ruled against India in retrospective tax dispute between the Indian Government and... View Article

State of World Population Report (SoWP) [Latest Report 2023]

The State of World Population is an annual report published by the United Nations Population Fund (UNFPA). Each edition covers and analyses... View Article

Atlantic Meridional Overturning Circulation (AMOC)

The Atlantic Meridional Overturning Circulation (AMOC) is a large system of ocean currents that carry warm water from the tropics northwards into... View Article

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