Rajasthan Board RBSE Class 12 Maths Chapter 9 समाकलन Miscellaneous Exercise

निम्नलिखित के मान ज्ञात कीजिए

प्रश्न 1.
∫[1 + 2 tan x (tan x + see x) dx
हल :
∫[1 + 2 tan x (tan x + sec x)] dx
= ∫(1 + 2 tan² x + 2 tan x sec x] dx
= ∫[2(1 + tan² x) + 2 sec x tan x – 1] dx
= 2∫(sec² x + sec x tan x) dx – ∫dx
= 2(tan x + sec x) – x + C

प्रश्न 2.
∫ex sin3 x dx
हल :
∫ex sin3 x dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 3.
∫ x² log (1 – x²)dx
हल :
∫ x² log (1 – x²)dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 4.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
यही अभीष्ट हल है।

प्रश्न 5.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 6.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
= [x tan x – ∫tan x dx] – [xsec x – ∫sec x dx]
= [x tan x – log| sec x |] – [xsec x – log| sec x + tan x |] + C
= x tan x – log| sec x | – x sec x + log x + tan x + C
= x | tan x – sec x | – log | sec x | + log | sec x + tan x | + C

प्रश्न 7.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 8.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 9.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 10.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 11.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 12.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
= tan-1 u + c = tan-1 t² + C
= tan-1 (tan² t) + C

प्रश्न 13.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 14.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
= ∫secx² dx + ∫cosec² xdu – 3 ∫dx
= tan x – cotx – 3x + C

प्रश्न 15.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 16.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
माना tan x + 2 = y
sec²x dx = dy
tan x + 2
tan x = y – 2
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 17.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 18.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
माना log x = t
\frac { 1 }{ x } dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
1 = A(3t + 2) + B(2t + 1)
1 = (3A + 2B)t + (2A + B)
तुलना करने पर,
3A + 2B = 0 तथा 2A + B = 1
हल करने पर,
A = 2, B = – 3
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 19.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
माना sec²2x = t
4 sin 2x.cos 2x dx = dt
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 20.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
माना sin – cos x = t
(sin x + cos x) dx = dt
⇒ (sin x – cos x)² = t²
⇒ sin² x + cos² x – 2 sin x cos x = t²
⇒ 1 – sin 2x = t²
⇒ sin 2x = 1 – t²
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 21.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions

प्रश्न 22.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
का मान है
(a) tan x + x + C
(b) cot x + x + C
(c) tan x – x + C
(d) cot x – x + C
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
= ∫sec² x dx – ∫1 dx
= tan x – x + C
अत: विकल्प (c) सही है।

प्रश्न 23.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
का मान है
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
अतः विकल्प (b) सही है।

प्रश्न 24.
∫log x dx बराबर है
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
∫log x dx
= ∫log x.1 dx
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
= x log x – x loge + C
= x log \frac { x }{ e } + c
अत: विकल्प (c) सही है।

प्रश्न 25.
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
बराबर है
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
हल :
माना है।
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
⇒ 1 = A(x + 1) + B(x)
⇒ 1 = (A + B)x + A
तुलना से, A = 1, A + B = 0
हुल करने पर,
A = 1, B = -1
RBSE Solutions for Class 12 Maths Chapter 9 समाकलन Additional Questions
अत: विकल्प (a) सही है।