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∫{e^x [sin x + cos x] / (1 – sin² x)} dx =

1) ex cosec x + c 2) ex + cot x + c 3) ex sec x + c 4)... View Article

∫{√[(e^x + a) / (e^x – a)]} dx =

1) \(\begin{array}{l}\cosh ^{-1}\left(\frac{e^{x}}{a}\right)+\sec ^{-1}\left(\frac{e^{x}}{a}\right)+C\end{array} \) 2) \(\begin{array}{l}\cosh ^{-1}\left(\frac{e^{x}}{a}\right)-\sec ^{-1}\left(\frac{e^{x}}{a}\right)+C\end{array} \) 3) \(\begin{array}{l}\frac{1}{a}\left[\cosh ^{-1}\left(\frac{e^{x}}{a}\right)+\sec ^{-1}\left(\frac{e^{x}}{a}\right)\right]+C\end{array} \) 4) \(\begin{array}{l}\frac{1}{a}\left[\cosh ^{-1}\left(\frac{e^{x}}{a}\right)-\sec ^{-1}\left(\frac{e^{x}}{a}\right)\right]+C\end{array} \)... View Article

∫{(1 + cot² x) / (1 + cot x)} dx =

1) – log |1 + cot x| + c 2) log |1 + tan x| + c 3) log (1... View Article

∫{sin 2x / [sin⁴ x + cos⁴ x]} dx =

1) 2 tan-1 (tan2 x) + c 2) tan-1 (x tan2 x) + c 3) tan-1 (tan2 x) + c... View Article

∫[2^x / (√(1 – 4^x))] dx =

1) (log 2) sin-1 2x + c 2) (1 / 2) sin-1 2x + c 3) (1 / log 2)... View Article

If {√(cot x) / [sin x cos x]} dx = P √cot x + Q, then the value of P is

1) -2 2) 2 3) 3 4) -3 Solution: (1) -2 \(\begin{array}{l}\begin{array}{l} \int \frac{\sqrt{\cot x}}{\sin x \cos x} d x=P... View Article

If f (x) = ∫dx / [(1 + x²)^3/2] and f (0) = 0, then the value of f (1) is

1) 1 / √2 2) – 1 / √2 3) √2 4) – √2 Solution: (1) 1 / √2 \(\begin{array}{l}\begin{array}{l}... View Article

If n ∈ N and I_n = ∫(log x)ⁿ dx, then I_n + nI_n-1 =

1) [log x]n+1 / (n + 1) 2) x (log x)n + c 3) (log x)n-1 4) [(log x)n /... View Article

∫{e^x [(x² + 1) / (x + 1)²]} dx =

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

∫{(2 dx) / [(e^x + e^-x)²]} =

1) [- e-x / (ex + e-x)] + c 2) [1 / (ex + e-x)] + c 3) [1 /... View Article

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 \(\begin{array}{l}\begin{array}{l} \int \frac{5 \tan x}{\tan x-2} d x \\... View Article

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

1) \(\begin{array}{l}\frac{-1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\}+C\end{array} \) 2) \(\begin{array}{l}\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\}+C\end{array} \) 3) \(\begin{array}{l}\frac{-1}{(\sec... 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... 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 /... View Article

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