Rajasthan Board RBSE Class 12 Maths Chapter 9 समाकलन Ex 9.2

निम्न फलनों को x के सापेक्ष समाकलन कीजिए

प्रश्न 1.
(a) ∫x sin x² dx
(b) ∫x√x² + 1 dx
हल :
(a)∫x sin x² dx
माना x² = t
⇒ 2x dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
(b) ∫x√x² + 1 dx
माना x² + 1 = t
⇒2x dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 2.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
(a)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना ex + cos x = t
⇒ (ex – sin x) = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
= log |t| + C
= log |ex + cos x|+C

(b)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना
1 + ex = t
ex dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 3.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
(a)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
(b)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 4.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 5.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 6.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
= log|cosec 2x – cot 2x| + log |cosec x – cot x |+ C

प्रश्न 7.
(a) ∫sin 3x sin 2x dx
(b) ∫ \sqrt { 1-sinx } dx
हल :
(a) ∫sin 3x sin 2x dx

RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 8.
(a) ∫cos4 x dx
(b) ∫sin3 x dx
हल :
(a) ∫cos x4 dx
= – ∫(cos² x)² dx

(b) ∫sin3 x dx

प्रश्न 9.

हल :

माना tan x = t
sec² x dx = dt

माना xex = t
⇒ (xex + ex.1)dx = dt
⇒ (1 + x)ex dx = dt

= ∫sec² t dt = tan t + c
= tan (xex) + c

प्रश्न 10.

हल :


माना cosx – sinx = t
⇒ (sin x + cos x) dx = dt

माना sin x + cos x = t
⇒ -cos x + sin x dx = dt

प्रश्न 11.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना tan x = t
⇒ sec²x dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना sin x + cos x = t
(cos x – sin x) dx = dt
= ∫ \frac { dt }{ t }
= log|t| + C
= log |sin x + cos x| + C

प्रश्न 12.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना x + a = t
dx = dt
x = t – a
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
= cos 2a ∫ dt – sin 2a ∫ cot dt
= (cos 2a)t – sin 2a log |sin t| + C1
= (x + a) cos 2a – sin 2a log |sin (x + a) | + C1
= x cos 2a – sin 2a log |sin(x + a)| + a cos 2a + C1
= x cos 2a – sin 2a log |sin (x + a)| + C
(जहाँ C = a cos 2a + C1)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना x – a = t
x = t + a
dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
= ∫cos a dt + ∫sin a cot dt
= cos a.t + sin a log |sin t|
= (x – a) cos a + sin a log |sin (x – a)| + C1
= x cos a + sin a log |sin (x – a)| + (-a cos a + C1)
= x cos a + sin a log |sin (x -a)| + C
(जहाँ C = -a cos a + C1)

प्रश्न 13.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 14.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना 4 = sin θ तथा 3 = r cos θ
तब्र r² sin² θ + r² cos² θ = 3² + 4² = 5²
⇒ r = 5
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 15.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
माना a cos² x + b sin² x = t
⇒ (-2a cos x sin x + 2b sin x cos x) dx = dt
⇒ (2b sin cos x – 2a sin x cos x) dx = dt
⇒ 2(b – a) sin x cos x dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

प्रश्न 16.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.2

= 2∫ (cos x + cos α) dx
= 2∫ cos x dx + 2∫ cos α dx
= 2 sin x + 2 cos α∫ dx
= 2 sin x + 2x cos α + C
= 2(sin x + x cos α) + C