Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.13 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

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

Question 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
Answer:
(1) ∫ (f(x)2 dx

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Explaination:
Given ∫ f (x) dx = g (x) + c
\(\frac{\mathrm{d}}{\mathrm{d} x}\) ∫ f(x)dx = \(\frac{\mathrm{d}}{\mathrm{d} x}\) (g(x) + c)
∫ \(\frac{\mathrm{d}}{\mathrm{d} x}\) (f(x)) dx = g'(x)
∫ d(f(x)) = g'(x)
f(x) = g(x)
∴ ∫ f(x) g'(x) dx = ∫ f(x) f(x) dx
= ∫ [f(x)]2 dx

Question 2.
If \(\int \frac{3^{\frac{1}{x}}}{x^{2}}\) dx = k\(\left(3^{\frac{1}{x}}\right)\) + c, then the value of k is
(1) log 3
(2) – log 3
(3) \(-\frac{1}{\log 3}\)
(4) \(\frac{1}{\log 3}\)
Answer:
(3) \(-\frac{1}{\log 3}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 1

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 3.
If ∫ f'(x) ex2 dx = (x – 1)ex2 + c, then f(x) is
(1) 2x3 – \(\frac{x^{2}}{2}\) + x + c
(2) \(\frac{x^{3}}{2}\) + 3x2 + 4x + c
(3) x3 + 4x2 + 6x + c
(4) \(\frac{2 x^{3}}{3}\) – x2 + x + c
Answer:
(4) \(\frac{2 x^{3}}{3}\) – x2 + x + c

Explaination:
Given ∫ f'(x) ex2 dx = (x – 1)ex2 + c
Differentiating both sides with respect to x we have
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 2

Question 4.
The gradient (slope) of a curve at any point (x, y) is \(\frac{x^{2}-4}{x^{2}}\). If the curve passes through the point (2, 7), then the equation of the curve is
(1) y = x + \(\frac{4}{x}\) + 3
(2) y = x + \(\frac{4}{x}\) + 4
(3)y = x2 + 3x + 4
(4) y = x2 – 3x + 6
Answer:
(1) y = x + \(\frac{4}{x}\) + 3

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 3
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 4
Given, this curve passes through the point (2, 7)
∴ 7 = 2 + \(\frac{4}{2}\) + c
7 = 2 + 2 + c
c = 7 – 4 = 3
∴ The required equation is
y = x + \(\frac{4}{x}\) + 3

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 5.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 5
(1) cot (xex) + c
(2) sec (xex) + c
(3) tan (xex) + c
(4) cos (xex) + c
Answer:
(3) tan (xex) + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 6
= ∫sec2 t
= tan t + c
= tan (x ex) + c

Question 6.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 7
(1) \(\sqrt{\tan x}+\mathbf{c}\)
(2) \(2 \sqrt{\tan x}+\mathbf{c}\)
(3) \(\frac{1}{2} \sqrt{\tan x}+c\)
(4) \(\frac{1}{4} \sqrt{\tan x}+c\)
Answer:
(1) \(\sqrt{\tan x}+\mathbf{c}\)

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 8

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 7.
∫sin3 dx is
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 9
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 10
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 11
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 12
Answer:
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 11

Explaination:
∫sin3 dx
sin 3x = 3 sin x – 4 sin3 x
4 sin3 x = 3 sin x – sin 3x
sin3 x = \(\frac{1}{4}\)(3 sin x – sin 3x)
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 13

Question 8.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 14
(1) x + c
(2) \(\frac{x^{3}}{3}\) + c
(3) \(\frac{3}{x^{3}}\) + c
(4) \(\frac{1}{x^{2}}\) + c
Answer:
(2) \(\frac{x^{3}}{3}\) + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 15
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 16

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 9.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 17
(1) tan-1 (sin x) + c
(2) 2 sin-1 (tan x) + c
(3) tan-1 (cos x) + c
(4) sin-1 (tan x) + c
Answer:
(4) sin-1 (tan x) + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 18

Question 10.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 19
(1) x2 + c
(2) 2x2 + c
(3) \(\frac{x^{2}}{2}\) + c
(4) – \(\frac{x^{2}}{2}\) + c
Answer:
(3) \(\frac{x^{2}}{2}\) + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 20

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 11.
∫23x+5 dx is
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 21
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 22
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 23
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 24
Answer:
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 24

Explaination:
∫23x+5 dx
Put 3x + 5 = t
3 dx = dt
dx = \(\frac{1}{3}\) dt
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 25
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 26

Question 12.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 27
(1) \(\frac{1}{2}\) sin 2x + c
(2) –\(\frac{1}{2}\) sin 2x + c
(3) \(\frac{1}{2}\) cos 2x + c
(4) –\(\frac{1}{2}\) cos 2x + c
Answer:
(2) –\(\frac{1}{2}\) sin 2x + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 28

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 13.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 29
(1) ex tan-1 (x + 1) + c
(2) tan-1 (ex) + c
(3) ex \(\frac{\left(\tan ^{-1} x\right)^{2}}{2}\) + c
(4) ex tan-1 x + c
Answer:
(4) ex tan-1 x + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 30

Question 14.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 31
(1) cot x + sin-1 x + c
(2) – cot x + tan-1x + c
(3) – tan x + cot-1 x + c
(4) – cot x – tan-1x + c
Answer:
(4) – cot x – tan-1x + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 32
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 33

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 15.
∫x2 cos x dx is
(1) x2 sin x + 2x cos x – 2 sin x + c
(2) x2 sin x – 2x cos x – 2 sin x + c
(3) – x2 sin x + 2x cos x + 2 sin x + c
(4) – x2 sin x – 2x cos x + 2 sin x + c
Answer:
(1) x2 sin x + 2x cos x – 2 sin x + c

Explaination:
\(\int x^{2} \cos x d x\)
By Bernoullis formula dv = cosxdx
u = x2 v = sinx
u’ = 2x v1 = -cos x
u” = 2 v2 = -sinx
= uv – u’v1 + u”v2
= x2sin x + 2x cos x – 2 sin x + c

Question 16.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 34
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 35
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 36
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 37
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 38
Answer:
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 35

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 39
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 40

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 17.
\(\int \frac{d x}{e^{x}-1}\) is
(1) log |ex| – log |ex – 1| + c
(2) log |ex| + log |ex – 1| + c
(3) log |ex – 1| – log |ex| + c
(4) log |ex + 1| – log |ex| + c
Answer:
(3) log |ex – 1| – log |ex| + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 41

Question 18.
∫e-4x cos x dx is
(1) \(\frac{e^{-4 x}}{17}\) [4 cos x – sin x] + c
(2) \(\frac{e^{-4 x}}{17}\) [- 4 cos x – sin x] + c
(3) \(\frac{e^{-4 x}}{17}\) [4 cos x + sin x] + c
(4) \(\frac{e^{-4 x}}{17}\) [- 4 cos x – sin x] + c
Answer:
(2) \(\frac{e^{-4 x}}{17}\) [- 4 cos x – sin x] + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 42

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 19.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 43
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 44
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 45
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 46
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 47
Answer:
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 47

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 48

Question 20.
∫e-7x sin 5x dx is
(1) \(\frac{e^{-7 x}}{74}\) [- 7 sin 5x – 5 cos 5x] + c
(2) \(\frac{e^{-7 x}}{74}\) [7 sin 5x + 5 cos 5x] + c
(3) \(\frac{e^{-7 x}}{74}\) [7 sin 5x – 5 cos 5x] + c
(4) \(\frac{e^{-7 x}}{74}\) [- 7 sin 5x + 5 cos 5x] + c
Answer:
(1) \(\frac{e^{-7 x}}{74}\) [- 7 sin 5x – 5 cos 5x] + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 49

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 21.
∫x2 ex/2 dx is
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 50
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 51
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 52
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 53
Answer:
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 52

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 54
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 55

Question 22.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 56
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 57
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 58
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 59
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 60
Answer:
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 60

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 61
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 62

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 23.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 63
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 64
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 65
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 66
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 67
Answer:
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 66

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 68
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 69

Question 24.
∫sin √x dx 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
Answer:
(1) 2(- √x cos √x + sin √x) + c

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 70
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 71

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13

Question 25.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 72
(1) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 73
(2) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 74
(3) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 75
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 76
Answer:
(4) Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 76

Explaination:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.13 77

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