Maths MCQs for Class 12 with Answers Chapter 7 Integrals

## Integrals Class 12 Maths MCQs Pdf

1. Given ∫ 2^{x} dx = f(x) + C, then f(x) is

**Answer/Explanation**

Answer: c

Explaination:

2.

(a) sin² x – cos² x + C

(b) -1

(c) tan x + cot x + C

(d) tan x – cot x + C

**Answer/Explanation**

Answer: d

Explaination:

3.

(a) 2(sin x + x cos θ) + C

(b) 2(sin x – x cos θ) + C

(c) 2(sin x + 2x cos θ) + C

(d) 2(sin x – 2x cos θ) + C

**Answer/Explanation**

Answer: a

Explaination:

4. ∫cot²x dx equals to

(a) cot x – x + C

(b) cot x + x + C

(c) -cot x + x + C

(d) -cot x – x + C

**Answer/Explanation**

Answer: d

Explaination: (d), ∫ (cosec²x -1)dx = -cot x – x + C

5.

(a) log |sin x + cos x|

(b) x

(c) log |x|

(d) -x

**Answer/Explanation**

Answer: d

Explaination:

6. If ∫ sec²(7 – 4x)dx = a tan (7 – 4x) + C, then value of a is

(a) 7

(b) -4

(c) 3

(d) \(-\frac{1}{4}\)

**Answer/Explanation**

Answer: d

Explaination:

(d), ∫sec²(7 – 4x)dx = \(\frac{\tan (7-4 x)}{-4}\) + C = –\(\frac{1}{4}\) tan (7 – 4x) + C.

7. The value of X for which

(a) 1

(b) log_{e}4

(c) loe_{4} e

(d) 4

**Answer/Explanation**

Answer: b

Explaination:

8.

**Answer/Explanation**

Answer: c

Explaination:

9.

then value of a is equal to

(a) 3

(b) 6

(c) 9

(d) 1

**Answer/Explanation**

Answer: c

Explaination:

10.

**Answer/Explanation**

Answer:

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11.

**Answer/Explanation**

Answer: c

Explaination:

12.

(a) I_{1} > I_{2}

(b) I_{2} > I_{1}

(c) I_{1} = I_{2}

(d) I_{1} > 2I_{2}

**Answer/Explanation**

Answer: b

Explaination:

13. If a is such that \(\int_{0}^{a} x d x\) ≤ a + 4, then

(a) 0 ≤ a ≤ 4

(b) -2 ≤ a ≤ 0

(c) a ≤ -2 or a ≤ 4

(d) -2 ≤ a ≤ 4

**Answer/Explanation**

Answer: d

Explaination:

(d), as \(\int_{0}^{a}\) x dx ≤ a + 4

⇒ \(\frac{a²}{2}\) ≤ a + 4

⇒ a² – 2a — 8 ≤ 0

⇒ (a – 1)² ≤ (3)²

⇒ -3 ≤ a – 1 ≤ 3

⇒ -2 ≤ a ≤ 4

14. If \(\frac{d}{dx}\) f(x) = g(x), then antiderivative of g(x) is ________ .

**Answer/Explanation**

Answer:

Explaination:

f(x), as \(\frac{d}{dx}\) f(x) = g(x)

⇒ ∫ g(x)dx = f(x).

15. Derivative of a function is unique but a function can have infinite antiderivatives. State true or false.

**Answer/Explanation**

Answer:

Explaination: True, as ∫ f(x)dx = g(x) + C, C is constant can take different values but \(\frac{d}{dx}\) [g(x) + C] =f(x) only

16.

**Answer/Explanation**

Answer:

Explaination: \(\frac{2}{3}\) ∫ cosec x . cot x dx = –\(\frac{2}{3}\) ∫ cosec x + C

17. Find ∫(ax + b)^{3}dx [AI 2011]

**Answer/Explanation**

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18. If ∫(ax + b)² dx = f(x) + C, find f(x)

**Answer/Explanation**

Answer:

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19. We have \(\frac{d}{dx}\)(3x² + sin x – e^{x}) = 6x + cos x -e^{x}. Represent the expression in the form of anti derivative.

**Answer/Explanation**

Answer:

Explaination:

\(\frac{d}{dx}\) (3x² + sin x – e^{x}) = 6x + cos x – e^{x}

⇒ ∫ (6x + cos x – e^{x}) = 3x² + sin x – e^{x}

20.

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21.

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22. Evaluate ∫ (sin x + cos x)² dx

**Answer/Explanation**

Answer:

Explaination:

∫ (sin x + cos x)² dx = ∫ (sin²x + cos²x + 2sin x cos x)dx

= ∫(1 + sin 2x)dx = x – \(\frac{\cos 2 x}{2}\) + C

23.

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24.

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25. Find ∫(e^{x log a} + e^{a log x} + e^{a log a})dx

**Answer/Explanation**

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26. Evaluate \(\int e^{\frac{1}{2} \log x} d x\).

**Answer/Explanation**

Answer:

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27.

(a) 3^{x} + x^{3} + C

(b) log |3^{x} + x^{3}| + C

(c) 3x²+ 3^{x} log_{e} 3 +C

(d) log |3x² + 3^{x} log_{e} 3| + C

**Answer/Explanation**

Answer: d

Explaination:

28.

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29.

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30.

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31. Find ∫ sec² (7 – x)dx

**Answer/Explanation**

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32. Find \(\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x\)

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33. Find ∫2x sin(x² + 1) dx

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34.

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35.

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36.

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37.

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38.

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39.

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40.

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41.

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42.

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43.

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44. Evaluate ∫ sec^{4} x tan x dx

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45.

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46.

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47. Find ∫ cot x . log(sin x) dx [NCERT]

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48.

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49. Find ∫(e^{x} + 3x)² (e^{x} + 3)dx

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50.

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51.

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52. Find ∫ (cosx – sinx)² dx

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53. Evaluate \(\int \sqrt{1+\sin \frac{x}{4}} d x\)

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54.

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55.

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56.

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57.

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58.

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59.

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60. ∫ e^{x} sec x(1 + tan x)dx = ________ + C.

**Answer/Explanation**

Answer:

Explaination:

e^{x} sec x, as ∫e^{x }(sec x + sec x tan x) dx,

i.e. f(x) = sec x

f'(x) = sec x tan x,

using formula ∫ e^{x }{f(x) + f'(x)}dx

= e^{x }f(x) + C

61. If \(\int_{-1}^{4}\) f(x) dx =4 and \(\int_{2}^{4}\) (3 – f(x))dx = 7, then the value of \(\int_{-2}^{-1}\) f(x) dx is ________ .

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62.

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63.

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64.

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65.

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66. If \(\int_{0}^{a}\) 3x² dx = 8 write the value of a. [Foreign 2017]

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67. Evaluate. \(\int_{2}^{3}\) 3^{x} dx [Delhi 2017]

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68.

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69. \(\int_{0}^{2a}\)f(x)dx = 2 \(\int_{0}^{a}\) f(x)dx if f(2a -x)= f(x). State true or false.

**Answer/Explanation**

Answer:

Explaination: True; result

70.

then value of a is ________ .

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71.

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72. \(\int_{-1}^{1}\) |(1 – x)| dx is equal to ________ .

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73.

is equal to 0.State true or false.

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74. The value of \(\int_{0}^{\pi}\) | cos x|dx is 2. State true or false.

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75. The value of \(\int_{-\pi}^{\pi}\) sin^{3}x cos²x dx is ________ .

**Answer/Explanation**

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Explaination: 0, as f(x) = sin^{3 }x. cos² x dx is an odd function

76.

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77.

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78.

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79.

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80. Evaluate \(\int_{-1}^{1}\) x|x|dx

**Answer/Explanation**

Answer:

Explaination:

Consider \(\int_{-1}^{1}\) x|x| dx

f(x) = x|x|, f(-x) = (-x)|-x| = -x|x| = -f(x)

Odd function.

∴ \(\int_{-1}^{1}\) x|x|dx = 0

[using \(\int_{-a}^{a}\) f(x) = 0, if f(x) is odd function]

81. Evaluate \(\int_{0}^{2\pi}\) cos^{5}x dx [Foreign 2017]

**Answer/Explanation**

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82.

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83. Evaluate \(\int_{0}^{1}\) [2x]dx

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84. Evaluate \(\int_{1}^{4}\) f(x) dx, where

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85.

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86. Evaluate \(\int_{-\pi}^{\pi}\) (sin^{-93} x + x^{295}) dx

**Answer/Explanation**

Answer:

Explaination:

\(\int_{-\pi}^{\pi}\) (sin^{-93} x + x^{295})dx,f(x) is odd function as f(-x) = -f(x)

∴ \(\int_{-\pi}^{\pi}\) (sin^{-93} x + x^{295}) dx=0

87.

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88.

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89.

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90. \(\int_{1}^{e}\) log x. dx

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91. Evaluate \(\int_{0}^{1}\) x(1 – x)^{89}dx

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92.

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93. Evaluate \(\int_{0}^{1}\) x²(1 -x)^{n}dx

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94.

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95. Evaluate \(\int_{0}^{\pi}\) |cos x|dx [DoE]

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