Exercise 11.13: Choose the correct answer

Integral Calculus (Mathematics)

Choose the correct or the most suitable answer from given four alternatives.

(1) If ∫ f (x)dx = g(x) + c, then ∫ f (x)g′(x)dx 

(1) ∫(f (x))² dx 

(2) ∫ f (x)g(x)dx 

(3) ∫ f ′(x)g(x)dx

(4) ∫(g(x))² dx

Ans: (1)

Solution

(2) If , then the value of k is 

(1) log 3 

(2) − log 3

(3) – 1 / log 3

(4) 1 / log 3

Ans: (3)

Solution

(3) If  ∫ f ′(x)e^x2dx = (x − 1)e^x2 + c , then f(x) is 

(1) 2x³ − x²/2 + x + c 

(2) x³/2 + 3x² + 4x + c 

(3) x³ + 4x² + 6x + c 

(4) 2x³/3 - x²+ x+ c

Ans: (4)

Solution

(4) The gradient (slope) of a curve at any point (x, y) is [x² – 4]/ x² . If the curve passes through the point (2, 7), then the equation of the curve is

(1) y = x + 4/x + 3 

(2) y = x + 4/x + 4 

(3) y = x² + 3x + 4 

(4) y = x² − 3x + 6

Ans: (1)

Solution

(5)  is 

(1) cot(xe^x ) + c 

(2) sec(xe^x ) + c 

(3) tan(xe^x ) + c 

(4) cos( xe^x ) + c

Ans: (3)

Solution

(6)  is 

(1) √tan x + c 

(2) 2√tan x + c 

(3) ½ √tan x + c 

(4) ¼ √tan x + c

Ans: (1)

Solution

(7) ∫sin³xdx is

Ans: (3)

Solution

(8)  is 

(1) x + c 

(2) [ x³/3] + c 

(3) [3/ x³] + c 

(4) [1/ x²] + c

Ans: (2)

Solution

(9)  is

 (1) tan^−1 (sin x ) + c 

(2) 2sin^−1 (tan x ) + c 

(3) tan^−1 (cos x ) + c 

(4) sin^−1 (tan x ) + c

Ans: (4)

Solution

(10)   is 

(1) x² + c 

(2) 2x² + c 

(3) x²/2 + c 

(4) − x²/2  + c

Ans: (3)

Solution

(11) ∫ 2^{3 x+5} dx is 

Ans: (4)

Solution

(12)  is 

Ans: (2)

Solution

(13)  is 

(1) e^x tan^−1 (x + 1) + c 

(2) tan^−1 (e^x ) + c 

(3) e^x [(tan^−1 x)²/2] + c 

(4) e^x tan^−1x + c

Ans: (4)

Solution

(14)  is 

(1) cot x + sin^−1 x + c 

(2) − cot x + tan^−1 x + c 

(3) − tan x + cot^−1 x + c 

(4) − cot x − tan^−1 x + c 

Ans: (4)

Solution

(15) ∫ x² cos x dx is 

(1) x² sin x + 2x cos x − 2sin x + c 

(2) x² sin x − 2x cos x − 2sin x + c 

(3) − x² sin x + 2x cos x + 2sin x + c 

(4) − x² sin x − 2x cos x + 2sin x + c 

Ans: (1)

Solution

(16)  is 

(1) √[1-x²] + sin^−1 x + c 

(2) sin^−1 x −√[1-x²] + c

(3) log | x + √[1-x²] | − √[1-x²] + c 

(4) √[1-x²] + log | x + √[1-x²] | +c

Ans: (2)

Solution

 (17)  is 

(1) log | e^x |- log | e^x - 1| + c 

(2) log | e^x | + log | e^x - 1| + c 

(3) log | e^x - 1| -  log | e^x | + c 

(4) log | e^x + 1| - log | e^x | + c 

Ans: (3)

Solution

(18) ∫e^−4x cos x dx is

Ans: (2)

Solution

(19) 

Ans: (4)

Solution

(20) ∫e ^-7x sin 5x dx is 

(1) e^−7x/74 [−7sin 5x − 5cos5x ] + c

(2) e^−7x/74 [7sin 5x + 5cos5x ] + c

(3) e^−7x /74 [7sin 5x − 5cos5x ] + c

(4) e^−7x /74 [-7sin 5x + 5cos5x ] + c

Ans: (1)

Solution

(21) ∫x²e^x/2dx is

(1) x²e^x/2 – 4xe^x/2 – 8e^x/2 + c

(2) 2x²e^x/2 – 8xe^x/2 – 16e^x/2 + c

(3) 2x²e^x/2 – 8xe^x/2 + 16e^x/2 + c

(4) [x²e^x/2 / 2] – [xe^x/2 /4] +[e^x/2 /8 ]+ c

Ans: (3)

Solution

(22) 

Ans: (4)

Solution

(23) 

Ans: (3)

Solution

(24) ∫sin √xdx is 

(1) 2(− √x cos√x + sin √x) + c 

(2) 2(− √x cos√x - sin √x) + c 

(3) 2(− √x sin√x - cos √x) + c 

(4) 2(− √x sin√x + cos √x) + c 

Ans: (1)

Solution

(25) ∫e^√x dx is

(1) 2 √x (1− e^√x) + c 

(2) 2 √x (e^√x − 1) + c

(3) 2e^√x (1− √x ) + c 

(4) 2e^√x (√x − 1) + c

Ans: (4)

Solution