Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.5 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.5

Integrate the following functions with respect to x

Question 1.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 1
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 2
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 3

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 2.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 4
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 5

Question 3.
(2x – 5) (3x + 4x)
Answer:
∫(2x – 5) (3x + 4x) dx
= ∫(72 x + 8x2 – 180 – 20x) dx
= ∫ 72 x dx + ∫82x2 dx – ∫180 dx – ∫2o x dx
= 72∫x dx + 8∫x2dx – 180 ∫dx – 20 ∫x dx
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 6

Question 4.
cot2 x + tan2 x
Answer:
∫cot2 x + tan2 x
= ∫(cosec2x – 1 + sec2x – 1) dx
= ∫(cosec2x + sec2x – 2) dx
= ∫cosec2x dx + ∫sec2x dx – ∫2 dx
= – cot x + tan x – 2x + c
= tan x – cot x – 2x + c

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 5.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 7
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 8
[cos 2x = cos2x – sin2x-1
cos 2x = 2 cos2x – 1]
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 9
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 10
= 2 ∫(cos x + cos α) dx
= 2 ∫ cos x dx + 2 ∫ cos α dx
= 2 ∫ cos x dx + 2 ∫ cos α ∫ dx
= 2 sin x + 2 cos α(x) + c
= 2 sin x + 2x cos α + c

Question 6.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 11
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 12
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 13

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 7.
\(\frac{3+4 \cos x}{\sin ^{2} x}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 14
= 3 ∫ cosec2 x dx + 4 ∫ cot x cosec x . dx
= 3 ∫ cosec2 x dx + 4 ∫ cosec x cot x dx
= 3 × – cot x + 4 × – cosec x + c
= – 3 cot x – 4 cosec x + c

Question 8.
\(\frac{\sin ^{2} x}{1+\cos x}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 15
= ∫ (1 – cos x) dx
= ∫ dx – ∫ cos x dx
= x – sin x + cannot

Question 9.
\(\frac{\sin 4 x}{\sin x}\)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 16
[sin 2A = 2 sin A cos A]
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 17
= 4 ∫ cos 2x cos x . dx
= 2 ∫ 2 cos 2x cosx . dx
= 2 ∫ [cos(2x + x) + cos(2x – x)] dx
[2 cos A cos B = cos (A + B) + cos (A – B)]
= 2 ∫ (cos 3x + cos x) dx
= 2 ∫ cos 3x dx + 2 ∫ cos x dx
= 2 \(\frac{\sin 3 x}{3}\) + 2 sin x + c
= 2 [\(\frac{\sin 3 x}{3}\) + sin x] + c

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 10.
cos 3x cos 2x
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 18

Question 11.
sin2 5x
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 19

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 12.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 20
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 21
[sin 2A = 2 sin A cos A]
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 22

Question 13.
ex log a ex
Answer:
∫ex log a ex = ∫e log ax . ex . dx
= ∫ ax ex . dx
= ∫ (ae)x . dx
[∫ax . dx = \(\frac{\mathrm{a}^{x}}{\log \mathrm{a}}\) + c]
= \(\frac{(\mathrm{ae})^{x}}{\log (\mathrm{ae})}\) + c

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 14.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 23
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 24
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 25

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 15.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 26
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 27
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 28

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 16.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 29
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 30
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 31

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 17.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 32
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 33
Put x = – 3
– 3 + 1 = A (- 3 + 3) + B (- 3 + 2)
– 2 = A × 0 + B (- 1)
B = 2

Put x = – 2
– 2 + 1 = A(- 2 + 3) + B(- 2 + 2)
– 1 = A × 1 + B × 0
A = – 1
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 34
= – log |x + 2| + 2 log |x + 3| + c
= 2 log |x + 3| – log |x + 2|+ c

Question 18.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 35
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 36
1 = A(x + 2)2 + B (x – 1) (x + 2) + C (x – 1)

Put x= -2
1 = A(- 2 + 2)2 + B(- 2 – 1) (- 2 + 2) + C(- 2 – 1)
1 = A × 0 + B × 0 + C × – 3
C = \(\frac{1}{3}\)

Put x = 1
1 = A(1 + 2)2 + B (1 – 1) (1 + 2) + C(1 – 1)
1 = A × 32 + B × 0 + C × 0
A = \(\frac{1}{9}\)

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Put x = 0
1 = A (0 + 2)2 + B (0 – 1) (0 + 2) + C (0 – 1)
1 = A × 4 – 2B – C
1 = \(\frac{1}{9}\) × 4 – 2B + \(\frac{1}{3}\)
1 = \(\frac{4}{9}+\frac{1}{3}\) – 2BSamacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 37
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 38

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 19.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 39
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 40
3x – 9 = A(x + 2) (x2 + 1) + B(x – 1) (x2 + 1) + (Cx + D) (x – 1) (x + 2)
3x – 9 = A (x + 2) (x2 + 1) + B(x – 1) + (x2 + 1) + Cx (x – 1) (x + 2) + D (x – 1) (x + 2)

Put x = – 2
3 × – 2 – 9 = A (- 2 + 2) ((2)2 + 1) + B(- 2 – 1) ((- 2)2 + 1) + C(- 2) (- 2 – 1) (- 2 + 2) + D(- 2 – 1) (- 2 + 2)
– 6 – 9 = A × 0 + B × (-3) (4 + 1) + C × 0 + D × 0
-15 = B ×- 3 × 5
-15 = – 15B ⇒ B = 1

Put x = 1
3 × 1 – 9 = A(1 + 2) (12 + 1) + B(1 – 1) (12 + 1) + C × 1 (1 – 1) (1 + 2) + D(1 – 1) (1 + 2)
3 – 9 = A × 3 × 2 + B × 0 + C × 0 + D × 0
– 6 = 6A ⇒ A = – 1

Put x = 0
3 × 0 – 9 = A(0 + 2) (02 + 1) + B(0 – 1) (02 + 1) + C × 0 (0 – 1) (0 + 2) + D(0 – 1) (0 + 2)
– 9 = 2A – B + 0 – 2D
– 9 = 2A – B – 2D
– 9 = – 2 × – 1 – 1 – 2D
– 9 = – 2 – 1 – 2D
9 = 3 + 2D
⇒ 2D = 9 – 3
⇒ 2D = 6 ⇒ D = 3

Put x = – 1
3 × – 1 – 9 = A(- 1 + 2) ((1)2 + 1) + B(- 1 – 1) ((- 1)2 + 1)) + C × – 1 × (- 1 – 1) (- 1 + 2) + D(- 1 – 1) (- 1 + 2)
– 3 – 9 = A × 1(1 + 1)+ B × (- 2) (1 + 1) – C × – 2 + D × – 2 × 1
– 12 = 2A – 4B + 2C – 2D
– 12 = 2 × -1 – 4 × 1 + 2C – 2 × 3
– 12 = – 2 – 4 + 2C – 6
– 12 = – 12 + 2C ⇒ C = 0
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 41
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 42

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Question 20.
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 43
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 44
Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 45
1 = A(x – 2) + B(x – 1)

put x = 2
1 = A(2 – 2) + B(2 – 1)
1 = A × 0 + B × 1
B = 1

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5

Put x = 1
1 = A(1 – 2) + B(1 – 1)
1 = A × – 1 + B × 0
A = – 1

Samacheer Kalvi 11th Maths Guide Chapter 11 Integral Calculus Ex 11.5 46