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∫{(e^x – 1) / (e^x + 1)} dx = f (x) + c, then f (x) =

1) 2 log (ex + 1) 2) 2 log |e2x - 1| 3) 2 log |ex + 1| - x 4) 2 log (e2x + 1) Solution: (3) 2 log |ex + 1| - x... View Article

∫dx / (cos x – sin x) =

The value of [latex]\int e^{\tan ^{-1}} \frac{\left(1+x+x^{2}\right)}{1+x^{2}} d x[/latex] is

1) tan-1 x + c 2) etan^(-1)x + 2x + c 3) etan^(-1)x + c 4) etan^(-1)x-x + c 5) xetan^(-1)x + c Solution: (5) xetan^(-1)x + c... View Article

∫(4^x+1 – 7^x-1) / 28^x =

1) (1 / (7 loge 4) (4-x)) - (4 / (loge 7) (7-x) + c 2) (1 / (7 loge 4) (4-x)) + (4 / (loge 7) (7-x) + c 3) (1 / (loge 7)) - ((7-x) / (loge 4) +... View Article

∫(sin⁶ x + cos⁶ x + 3 sin² x cos² x) dx =

1) x + c 2) (3 / 2) sin 2x + c 3) (- 3 / 2) cos 2x + c 4) (1 / 3) sin 3x - cos 3x + c 5) (1 / 3) sin 3x + cos 3x + c Solution: (1) x + c... View Article

The value of ∫{(x² + 1) / (x² – 1)} dx is

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

∫dx / (x² + 4x + 13) =

1) log (x2 + 4x + 3) + c 2) (1 / 3) tan-1 [(x + 2) / 3] + c 3) log (2x + 4) + c 4) {[2x + 4] / [x2 + 4x + 13]2} + c Solution: (2) (1 / 3)... View Article

∫dx / (cos x + √3 sin x) =

Let f (x) be an indefinite integral of sin² x. Statement I: The function F (x) satisfies F (x + π) = F (x) for all real x. Statement II: sin² (x + π) = sin² x for all real x.

1) Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I 2) Statement I is correct, Statement... View Article

If ∫dx / (x² + 2x + 2) = f (x) + c, then f (x) =

1) tan-1 (x + 1) 2) 2 tan-1 (x + 1) 3) - tan-1 (x + 1) 4) 3 tan-1 (x + 1) Solution: (1) tan-1 (x + 1) ∫dx / (x2 + 2x + 2) = ∫dx / (x + 1)2 +... View Article

∫{(x³ + 3x² + 3x + 1) / (x + 1)⁵} dx =

1) (- 1 / (x + 1)) + c 2) (1 / 5) log (x + 1) + c 3) log (x + 1) + c 4) tan-1 x + c Solution: (1) (- 1 / (x + 1)) + c ∫{(x3 + 3x2 + 3x + 1)... View Article

∫cosec⁴ x dx =

1) cot x + (cot3 x / 3) + c 2) tan x + (tan3 x / 3) + c 3) (- cot x) - (cot3 x / 3) + c 4) (- tan x) - (tan3 x / 3) + c Solution: (3) (- cot... View Article

The value of [latex]int frac{1+x+sqrt{x+x^{2}}}{sqrt{x}+sqrt{1+x}} dx = [/latex]

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

∫e^3logx (x⁴ + 1)^-1 dx =

1) e3logx + c 2) (1 / 4) log |x4 + 1| + c 3) log (x4 + 1) + c 4) (1 / 2) log (x4 + 1) + c 5) (x4) / (x4 + 1) + c Solution: (2) (1 / 4) log... View Article

∫{(sec x cosec x) / (2 cot x – sec x cosec x)} dx =

∫[(sec x) / (sec x + tan x)] dx =

1) tan x - sec x + C 2) log (1 + sec x) + C 3) secx + tan x+ C 4) log sin x + log cos x + C Solution: (1) tan x - sec x + C ∫[(sec x) / (sec... View Article

If f (x) = lim_y→x [sin² y – sin² x] / [y² – x²], then ∫4x f (x) dx =

1) cos 2x + C 2) 2 cos 2x + C 3) - cos 2x + C 4) - 2 cos 2x + C Solution: (3) - cos 2x + C... View Article

∫(sec² x – 7) / (sin⁷ x) dx =

1) [tan x / sin7 x] + c 2) [cos x / sin7 x] + c 3) [sin x / cos7 x] + c 4) [sin x / tan7 x] + c Solution: (1) [tan x / sin7 x] + c ∫(sec2 x... View Article

∫dx / [√(1 – x) (x – 2)] =

1) sin-1 (2x - 3) + C 2) sin-1 (2x + 5) + C 3) sin-1 (3 - 2x) + C 4) sin-1 (5 - 2x) + C Solution: (1) sin-1 (2x - 3) + C... View Article

∫[e^alogx + e^xloga] dx = ….. + c, a, x > 1

1) [xa+1 / (a + 1)] + (ax / log a) 2) [ealogx / a logx] + [exloga / x log a] 3) [xa-1 / a - 1] + ax . loga + c 4) [ealogx / (a / x)] +... View Article

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