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The value of ∫₁² dx / [x (1 + x⁴)] is

1) (1 / 4) log (17 / 32) 2) (1 / 4) log (32 / 17) 3) log (17 /... View Article

The value of ∫₀¹ [√(1 – x) / (1 + x)] dx is

1) (π / 2) + 1 2) (π / 2) – 1 3) -1 4) 1 Solution: (2) (π / 2) –... View Article

The value of ∫₁⁴ e^√x dx is

1) e2 2) 2e2 3) 4e2 4) 3e2 Solution: (2) 2e2 \(\begin{array}{l}\begin{array}{l} I=\int_{1}^{4} e^{\sqrt{x}} d x \\ \text { Let... View Article

The value of ∫₂³ [x + 1] / [x² (x – 1)] dx is

1) [log (16 / 9) + (1 / 6)] 2) [log (16 / 9) – (1 / 6)] 3) 2... View Article

∫₀^∞ dx / [(x + √x² + 1)³] =

1) 3/8 2) 1/8 3) -3/8 4) None of these Solution: (1) ⅜ \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\infty} \frac{d x}{\left(x+\sqrt{x^{2}+1}\right)^{3}} \\ \text {... View Article

The value of ∫_-π/4^π/4 sin^-4 x dx is

1) – (8/3) 2) 3/2 3) 8/3 4) None of these Solution: (1) – (8/3) \(\begin{array}{l}\begin{array}{l} I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{-4} x... View Article

If k ∫₀¹ x . f (3x) dx = ∫₀³ t . f (t) dt, then the value of k is

1) 9 2) 3 3) 1/9 4) 1/3 Solution: (1) 9 \(\begin{array}{l}\begin{array}{l} k \int_{0}^{1} x f(3 x) d x \\... View Article

The value of ∫₁² dx / [(x + 1) (√x² – 1)] is

1) 1 2) √3 3) 2/√3 4) -2/√3 Solution: (2) √3 \(\begin{array}{l}\begin{array}{l} I=\int_{1}^{2} \frac{d x}{(x+1) \sqrt{x^{2}-1}} \\ \text { Let... View Article

The value of ∫_π/4^π/2 e^x (log sin x + cot x) dx =

1) eπ/4 log 2 2) – eπ/4 log 2 3) (1 / 2) eπ/4 log 2 4) (- 1 /... View Article

The value of ∫₀^a [√(a – x) / x] dx is

1) a/2 2) a/4 3) πa/2 4) πa/4 Solution: (3) πa/2 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{a} \sqrt{\frac{a-x}{x}} d x \\ \text { Let... View Article

∫₀¹ 1 / (x + √x) dx =

1) log 3 2) log 1 3) log 4 4) log 2 Solution: (3) log 4 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{1} \frac{d x}{x+\sqrt{x}}... View Article

∫₀^π 1 / [1 + sin x] dx =

1) 1 2) 2 3) -1 4) -2 Solution: (2) 2 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\pi} \frac{1}{1+\sin x} d x \\ =\int_{0}^{\frac{\pi}{2}} \frac{1}{\left(\cos... View Article

The value of ∫₀^∞ dx / [(x² + 4) (x² + 9)] is

1) π/60 2) π/20 3) π/40 4) π/80 Solution: (1) π/60 \(\begin{array}{l}\begin{array}{l} \int_{0}^{\infty} \frac{d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)} \\ =\frac{1}{5}\left[\int_{0}^{\infty} \frac{1}{\left(x^{2}+4\right)} d x-\int_{0}^{\infty}... View Article

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to

a) 12 b) -12 c) -24 d) 6 Solution: p’(1) = 0 and p’(2) = 0 p’(x) = a(x-1)(x-2) p(x)... View Article

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