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∫_-π/3^π/3 [x sin x] / [cos² x] dx =

1) (1 / 3) (4π + 1) 2) [(4π / 3) – 2 log tan (5π / 12)] 3) [(4π /... View Article

∫₀^π/2 x sin² x cos² x dx =

1) π2/32 2) π2/16 3) π/32 4) None of these Solution: (4) None of these \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\frac{\pi}{2}} x \sin ^{2}... View Article

∫₀^π/2 (√cot x) / [(√cot x + √tan x)] dx =

1) 1 2) -1 3) π/2 4) π/4 Solution: (4) π/4 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} d x \ldots(i)\\... View Article

If f (x) = x – [x], for every x, where [x] is the greatest integer less than or equation to x. Then, ∫_-1¹ f (x) dx =

1) 1 2) 2 3) 3 4) 0 5) 1/2 Solution: (1) 1 \(\begin{array}{l}I =\int_{-1}^{1} f(x) d x \\ =\int_{-1}^{1}(x-[x])... View Article

∫_-10¹⁰ log [(a + x) / (a – x)] dx =

1) 0 2) -2 log(a + 10) 3) 2 log [(a + 10) / (a – 10)] 4) 2 log(a... View Article

∫₀² [x²] dx =

1) 2 – √2 2) 2 + √2 3) √2 – 1 4) -√2 – √3 5) None of these... View Article

∫₀^π |cos x| dx =

1) 1/2 2) -2 3) 1 4) -1 5) 2 Solution: (5) 2 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\pi}|\cos x| d x=\int_{0}^{\frac{\pi}{2}} \cos x-\int_{\frac{\pi}{2}}^{\pi}... View Article

The value of ∫_-π^π sin³ x cos² x dx is

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

The value of ∫_π^2π [2 sin x] dx is

1) π/3 2) – (5π/3) 3) 4π/3 4) – (π/3) Solution: (2) – (5π/3) y = [2 sin x] in... View Article

∫₀^π/4 log (1 + tan x) dx =

1) (π / 8) loge 2 2) (π / 4) log2 e 3) (π / 4) loge 2 4) (π... View Article

The value of ∫₃⁶ [√x / [√(9 – x) + √x]] dx is

1) 3/2 2) 2 3) 1 4) 2 Solution: (1) 3/2 \(\begin{array}{l}I =\int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x \quad \ldots(1) \\ I... View Article

If f (x) = f (a – x) and g (x) + g (a – x) = 2, then the value of ∫₀^a f (x) g (x) dx is

1) ∫0a f (x) dx 2) ∫0a g (x) dx 3) ∫0a [g (x) – f (x)] dx 4) ∫0a... View Article

∫₀^π cos³ x dx =

1) 0 2) 1 3) -1 4) 1/2√2 Solution: (1) 0 \(\begin{array}{l}I=\int_{0}^{\pi} \cos ^{3} x d x \ldots(i) \\ x... View Article

The value of ∫_-π/2^π/2 √cos x – cos³ x dx is

1) 0 2) 4/3 3) 2/3 4) 1/5 Solution: (2) 4/3 \(\begin{array}{l}I =\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^{3} x} d x \\... View Article

∫₀^2π sin⁶ x cos⁵ x dx =

1) 2π 2) π/2 3) 0 4) -π Solution: (3) 0 \(\begin{array}{l}I =\int_{0}^{2 \pi} \sin ^{6} x \cos ^{5} x... View Article

The value of I = ∫₀¹ x |x – (1 / 2)| dx is

1) 1/3 2) 1/4 3) 1/8 4) None of these Solution: (3) ⅛ \(\begin{array}{l}I =\int_{0}^{1} x\left|x-\frac{1}{2}\right| d x \\ =\int_{0}^{\frac{1}{2}}... View Article

The value of ∫₀^π/2 (sin¹⁰⁰ x – cos¹⁰⁰ x) dx is

1) 1/100 2) 100!/(100)100 3) π/100 4) 0 Solution: (4) 0 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{\frac{\pi}{2}}\left(\sin ^{100} x-\cos ^{100} x\right) d x \\... View Article

The value of integral from minus pi to pi fraction cos2x by 1+ax dx where a greater than zero is

(a) 2π (b) π/a (c) π/2 (d) aπ Solution: (c) π/2 \(\begin{array}{l}I= \int_{-\pi}^{\pi} \frac{\cos ^{2} x}{1+a^{x}} d x \ldots .(i)... View Article

The value of ∫₀¹ tan^-1 [(2x – 1) / (1 + x – x²)] dx is

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

∫₀¹⁰ |x (x – 1) (x – 2)| dx =

1) 160.05 2) 1600.5 3) 16.005 4) None of these Solution: (2) 1600.5 \(\begin{array}{l}\begin{array}{l} I=\int_{0}^{10}|x(x-1)(x-2)| d x \\ =\int_{0}^{1}+(x)(-(x-1))(-(x-2)) d... View Article

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