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RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

May 16, 2019 by Prasanna Leave a Comment

Rajasthan Board RBSE Class 12 Maths Chapter 9 समाकलन Ex 9.6

निम्न फलनों का x के सापेक्ष समाकलन कीजिए

प्रश्न 1.
(a) ∫x cos
(b) ∫x sec²x
हल :
(a) ∫x cos dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= x sin x – ∫1.sin x dx
= x sin x – ( – cos x) + C
= x sin x + cos x + C

(b) ∫x sec²x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= x tan x – ∫1.tan x dx
= x tan x + log |cos x| + C
= x tan x – log |sec x| + C

प्रश्न 2.
(a) x3e-x
(b) x3 sinx
हल :
(a) ∫x3e-x
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= – x3e-x + 3∫x2e-x dx ….(i)
∫x² e-x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= – x²e-x + ∫2xe-x dx
= – x²e-x + 2∫xe-x dx ……(ii)
∫x e-x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= – xe-x + ∫e-x dx
= – xe-x – e-x + C …..(iii)
समी. (i), (ii) व (iii) को हल करने पर
∫x3e-x dx = – x3e-x + 3[- x2e-x + 2 ∫xe-x dx]
= -x3e-x + 3(-x2e-x + 2(-xe-x – e-x + C1))
= -x3e-x – 3x2e-x + 6(-xe-x – e-x + C1)
= -x3e-x – 3x2e-x – 6xe-x – 6e-x + 6C1
= -x3e-x – 3x2e-x – 6xe-x – 6e-x + C
= -e [x3 + 3x2 + 6x + 6] + C

(b) ∫x3 sinx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= -x3 cos x + 3x2 sin x – 6(-x cos x + ∫cos x dx)
= -x3 cosx + 3x2 sin x + 6x cos x – 6 sin x + C

प्रश्न 3.
(a) ∫x3 (log x)² dx
(b) ∫x3ex2 dx.
हल :
(a) ∫x3 (log x)² dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

(b) ∫x3ex2 dx.
माना x² = t
⇒ 2x dx = dt
⇒ x dx = \frac { dt }{ 2 }
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 4.
(a) ∫ e2x eex dx
(b) ∫ (log x)² dx
हल :
(a) ∫ e2x eex dx
माना ex = t
ex dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= t et – ∫1 et dt = tet – et + c
= et (t – 1) + c
= eex (ex – 1) + c

(b) I = ∫ (log x)² dx = ∫(log x)².1 dx
[(log x)² को प्रथम फलन तथा 1 को द्वितीय फलन लेने पर ]
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
(log x को प्रथम फलन तथा 1 को द्वितीय फलन लेने पर)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= x (log x)² – 2x log x + 2x + C

प्रश्न 5.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
(a) I = ∫ cos-1x dx
माना cos-1 x = t
तब x = cos t
dx = – sint dt
∴ I = – ∫t sin t dt
(t को प्रथम फलन तथा sin t को द्वितीय फलन लेने पर)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
माना x = a tan²θ
∴ dx = 2a tan θ sec² θ dθ
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
(θ को पहला एवं tan θ sec² θ को दूसरा फलन मानने पर)
खण्डशः समाकलन से,
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 6.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
(a) I = ∫ sin-1 (3x – 4x3) dx
माना x = sin t, तब dx = cos t dt
∴ I = ∫sin-1(3 sint – 4 sin3 t).cos t dt
= ∫sin-1 (sin 3t) cos t dt
= ∫3t cos t dt
= 3∫t cost dt
(t को प्रथम फलन तथा cos t को द्वितीय फलन लेने पर)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
(x को प्रथम फलन तथा sec² \frac { x }{ 2 } को द्वितीय फलन लेने पर)।
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 7.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
(a) माना x = cos θ
dx = – sinθ dθ,
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

(b) ∫cos √x dx
माना √x = t, x = t²
dx = 2t dt
∫cos √x dx = ∫cos t.2t dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 8.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= ∫x sec² x dx – ∫x tan x sec x dx
= I1 – I2 …….(i)
I1 = ∫x sec² x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= x tan x – ∫1.tan x dx
= x tan x – ∫tan x dx
= x tanx – (- log |cos x|) + C1
= x tan x + log |cos x| + C1 ….(ii)
I2 = ∫x.tan x sec x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= x sec x – ∫1.sec dx
= x sec x – ∫sec x dx
= x sec x – log | sec x + tan x | + C2 …(iii)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 9.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 10.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= θ( – cos θ) + ∫ cos θ dθ
= – θ cos θ + sin θ + c
= sin θ – θ cos θ + c
= cos θ [tanθ – θ] + c
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 11.
∫ ex (cot x + log sin x) dx
हल :
∫ ex (cot x + log sin x) dx
माना I = ∫ex [log |sin x| + cot x]dx
= ∫ex log | sinx | dx + ∫ex cot x dx
= ∫log | sin x | ex dx + ∫ex cot x dx
अब ∫log |sin x| ex dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
∴ I = ex log | sin x |ex – ∫ex cotx dx + ∫ex cotx dx
= log | sin x |ex + C
= ex log | sin x | + C

प्रश्न 12.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= ∫x sec² x dx + ∫tan x dx
= x tan x – ∫tan x dx + ∫tan x dx
= x tan + C

प्रश्न 13.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
(केवल पहले भाग का खण्डशः समाकलन करने पर)
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 14.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 15.
∫ex [log (sec x + tan x) + sec x) dx
हल :
= ∫ex [log (sec x + tan x) + sec x) dx
= ∫ex log (sec x + tan x) dx + ∫ex sec x dx
= ∫log (sec x + tan x) ex dx + ∫sec xex dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= ex log |sec x + tan x| – ∫sec xex dx + ∫sec xex dx + C
= ex log |sec x + tan x| + C

प्रश्न 16.
∫ex (sin x + cos x) sec² x dx
हल :
∫ex (sin x + cos x) sec² x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
= ∫ex (sec x + sec x tan) dx
= ex sec x + C [∵ ex [f(x) + f'(x)] dx = ex f(x) + c , f(x) = sec x , f'(x) = sec x tan x]

प्रश्न 17.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 18.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 19.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 20.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 21.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6
Sec-1 x को प्रथम तथा 1 को द्वितीय फलन लेकर खण्डशः माकलन करने पर,
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

प्रश्न 22.
∫ (sin-1 x)² dx
हल :
माना I = ∫(sin-1 x)² dx
माना sin-1 x = θ
x = sinθ
∴ dx = c0s θ dθ
∴ I = ∫θ² • cos θ aθ
(θ² को प्रथम फलन एवं cos θ को द्वितीय फलन लेने पर)
= θ² sin θ – ∫2θ sin θ dθ + C
= θ² sin θ – 2∫θ sin θ + C
= θ² sin θ – 2[θ – cos θ) – ∫1.(-cos θ) dθ]+C
(पुन: θ को प्रथम तथा sin θ को द्वितीय फलन लेने पर)।
= θ² sin θ + 2θ cos θ – 2∫cos θ dθ+C
= θ² sin θ + 2θ cos θ – 2 sin θ + C
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Ex 9.6

RBSE Solutions for Class 12 Maths

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