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If ∫(x² / √1 – x) dx = p √(1 – x) [3x² + 4x + 8], then the value of p is

1) – 2 / 15 2) 2 / 15 3) 4 / 15 4) None of these Solution: (1) –... View Article

∫{dx / [x (x + 1)]} =

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

∫[(sin^-1 x) / (√1 – x²)] dx =

1) log (sin-1 x) + c 2) (1 / 2) (sin-1 x)2 + c 3) log (√1 – x2) +... View Article

∫(1 / x) log x dx =

1) loge (1 – loge x) + c 2) loge (loge ex – 1) + c 3) log (log x... View Article

∫{[cos x – sin x] / [1 + 2 sin x cos x]} dx =

1) [- 1 / (cos x – sin x)] + c 2) ([cos x + sin x] / [cos x... View Article

[latex]\int(\sqrt[3]{x})\left(\sqrt[5]{1+\sqrt[3]{x^{4}}}\right) d x=[/latex]

1) \(\begin{array}{l}\left(1+x^{3 / 4}\right)^{6 / 5}+C\end{array} \) 2) \(\begin{array}{l}\left(1+x^{4 / 3}\right)^{6 / 5}+C\end{array} \) 3) \(\begin{array}{l}\frac{5}{8}\left(1+x^{4 / 3}\right)^{6 / 5}+C\end{array}... View Article

[latex]\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} dx =[/latex]

1) \(\begin{array}{l}(1+x) e^{x+x^{-1}}+C\end{array} \) 2) \(\begin{array}{l}(x-1) e^{x+x^{-1}}+C\end{array} \) 3) \(\begin{array}{l}-x e^{x+x^{-1}}+C\end{array} \) 4) \(\begin{array}{l}x e^{x+x^{-1}}+C\end{array} \) Solution: (4) \(\begin{array}{l}x e^{x+x^{-1}}+C\end{array}... View Article

∫sin³ x . cos² x dx =

1) (sin5 x / 5) – (sin3 x / 3) + c 2) (sin5 x / 5) + (sin3 x... View Article

∫{1 / [1 + cos x + sin x]} dx =

1) (1 / 4) log |1 + tan (x / 2)| + c 2) (1 / 2) log |1 +... View Article

∫x log x dx =

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

∫[(1 + x⁴) / (1 – x⁴)^3/2] dx =

1) [x / √1 – x4] + c 2) [- x / √1 – x4] + c 3) [2x /... 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’... View Article

If ∫[(x² – 2) / (x³ √x² – 1))] dx =

1) [(x2) / √x2 – 1)] + c 2) [(- x2) / √x2 – 1)] + c 3) [(√x2 –... 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) ,... 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,... 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... View Article

∫[log (x + √1 + x²) / √1 + x²] dx =

1) [log (x + √1 + x2)]2 + c 2) x log (x + √1 + x2) + c 3)... 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 \(\begin{array}{l}\begin{aligned} \int \frac{(x+2) d x}{2 x^{2}+6 x+5}... View Article

∫[dx / (√1 – e^2x)] =

1) log |e-x + √e-2x – 1| + c 2) log |ex + √e2x – 1| + c 3) –... View Article

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

1) log |(sin x) / (1 + cos x)| + c 2) log |(sin x) / (x + cos x)|... View Article

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