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UP Board class 12 Maths (7. समाकलन) solution PDF
UP Board class 12 Maths 7. समाकलन is a Hindi Medium Solution which is prescribed by Uttar Pradesh Board for their students. These Solutions is completely prepared considering the latest syllabus and it covers every single topis, so that every student get organised and conceptual learning of the concepts. class 12 Students of UP Board who have selected hindi medium as their study medium they can use these Hindi medium textSolutions to prepare themselves for exam and learn the concept with ease.
UP Board class 12 Maths 7. समाकलन Hindi Medium Solutions - PDF
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अध्याय-7: समाकलन (Integrals)
महत्वपूर्ण सूत्र एवं परिभाषाएँ
| 1. ∫xn dx = (xn+1)/(n+1) + C, (n ≠ -1) | 2. ∫(1/x) dx = log |x| + C |
| 3. ∫ex dx = ex + C | 4. ∫ax dx = (ax)/(loge a) + C |
| 5. ∫sin x dx = -cos x + C | 6. ∫cos x dx = sin x + C |
| 7. ∫sec2 x dx = tan x + C | 8. ∫cosec2 x dx = -cot x + C |
| 9. ∫sec x tan x dx = sec x + C | 10. ∫cosec x cot x dx = -cosec x + C |
| 11. ∫(1/√(1-x2)) dx = sin-1 x + C | 12. ∫(1/(1+x2)) dx = tan-1 x + C |
| 13. ∫(1/(x√(x2-1))) dx = sec-1 x + C | 14. ∫tan x dx = log |sec x| + C |
| 15. ∫cot x dx = log |sin x| + C | 16. ∫sec x dx = log |sec x + tan x| + C |
| 17. ∫cosec x dx = log |cosec x - cot x| + C | 18. ∫eax sin bx dx = eax(a sin bx - b cos bx)/(a2+b2) + C |
| 19. ∫eax cos bx dx = eax(b sin bx + a cos bx)/(a2+b2) + C | |
प्रश्नावली 7.1
निम्नलिखित फलनों के प्रतिअवकलज (समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए:
प्रश्न 1. sin 2x
हल: हम जानते हैं कि d(cos 2x)/dx = -2 sin 2x
इसलिए, ∫sin 2x dx = -(1/2) cos 2x + C
प्रश्न 2. cos 3x
हल: हम जानते हैं कि d(sin 3x)/dx = 3 cos 3x
इसलिए, ∫cos 3x dx = (1/3) sin 3x + C
प्रश्न 3. e2x
हल: हम जानते हैं कि d(e2x)/dx = 2e2x
इसलिए, ∫e2x dx = (1/2)e2x + C
प्रश्न 4. (ax + b)2
हल: हम जानते हैं कि d(ax + b)3/dx = 3a(ax + b)2
इसलिए, ∫(ax + b)2 dx = (1/(3a))(ax + b)3 + C
प्रश्न 5. sin 2x - 4e3x
हल: ∫(sin 2x - 4e3x) dx = ∫sin 2x dx - 4∫e3x dx
= -(1/2) cos 2x - (4/3)e3x + C
UP Board समाधान - अभ्यास प्रश्न
निम्नलिखित समाकलनों को ज्ञात कीजिए:
प्रश्न 6. ∫(4e3x + 1) dx
हल: ∫(4e3x + 1) dx = 4∫e3x dx + ∫1 dx
= (4/3)e3x + x + C
प्रश्न 7. ∫x2(1 - 1/x2) dx
हल: ∫x2(1 - 1/x2) dx = ∫(x2 - 1) dx
= (x3/3) - x + C
प्रश्न 8. ∫(ax2 + bx + c) dx
हल: ∫(ax2 + bx + c) dx = a∫x2 dx + b∫x dx + c∫1 dx
= (a/3)x3 + (b/2)x2 + cx + C
प्रश्न 9. ∫(2x2 + ex) dx
हल: ∫(2x2 + ex) dx = 2∫x2 dx + ∫ex dx
= (2/3)x3 + ex + C
प्रश्न 10. ∫(√x - 1/√x)2 dx
हल: ∫(√x - 1/√x)2 dx = ∫(x - 2 + 1/x) dx
= (x2/2) - 2x + log|x| + C
विधि: प्रतिस्थापन द्वारा समाकलन
प्रश्न 11 से 37 तक के प्रश्नों में प्रतिस्थापन विधि का उपयोग करके समाकलन ज्ञात कीजिए।
प्रश्न 11. ∫(2x/(1+x2)) dx
हल: माना 1 + x2 = t, तब 2x dx = dt
∫(2x/(1+x2)) dx = ∫(1/t) dt = log|t| + C = log(1+x2) + C
प्रश्न 12. ∫(log x)2/x dx
हल: माना log x = t, तब (1/x) dx = dt
∫(log x)2/x dx = ∫t2 dt = t3/3 + C = (log x)3/3 + C
प्रश्न 13. ∫(1/(x(1+log x))) dx
हल: माना 1 + log x = t, तब (1/x) dx = dt
∫(1/(x(1+log x))) dx = ∫(1/t) dt = log|t| + C = log|1+log x| + C
खंडशः समाकलन
खंडशः समाकलन के सूत्र का उपयोग करें: ∫u v dx = u∫v dx - ∫[u'∫v dx] dx
प्रश्न 14. ∫x sin x dx
हल: u = x, v = sin x लेने पर
∫x sin x dx = x(-cos x) - ∫1(-cos x) dx
= -x cos x + sin x + C
प्रश्न 15. ∫x log x dx
हल: u = log x, v = x लेने पर
∫x log x dx = (log x)(x2/2) - ∫(1/x)(x2/2) dx
= (x2/2) log x - (1/2)∫x dx
= (x2/2) log x - (x2/4) + C
निश्चित समाकलन
निश्चित समाकलन के गुणधर्मों का उपयोग करते हुए निम्नलिखित समाकलनों का मान ज्ञात कीजिए:
प्रश्न 16. ∫0π/2 cos2 x dx
हल: cos2 x = (1+cos 2x)/2
∫0π/2 cos2 x dx = (1/2)∫0π/2 (1+cos 2x) dx
= (1/2)[x + (sin 2x)/2]0π/2
= (1/2)[(π/2 + 0) - (0 + 0)] = π/4
प्रश्न 17. ∫01 xex dx
हल: खंडशः समाकलन द्वारा:
∫01 xex dx = [xex]01 - ∫01 ex dx
= (1·e1 - 0) - [ex]01
= e - (e - 1) = 1
समाकलन के अनुप्रयोग
समाकलन का उपयोग करके क्षेत्रफल, आयतन आदि ज्ञात किए जा सकते हैं।
प्रश्न 18. वक्र y = x2, x-अक्ष एवं कोटियों x = 0 तथा x = 3 से घिरे क्षेत्र का क्षेत्रफल ज्ञात कीजिए।
हल: अभीष्ट क्षेत्रफल = ∫03 x2 dx
= [x3/3]03 = (27/3) - 0 = 9 वर्ग इकाई
प्रश्न 19. वक्र y = sin x, x-अक्ष एवं कोटियों x = 0 तथा x = π से घिरे क्षेत्र का क्षेत्रफल ज्ञात कीजिए।
हल: अभीष्ट क्षेत्रफल = ∫0π sin x dx
= [-cos x]0π = (-cos π) - (-cos 0)
= (-(-1)) - (-1) = 1 + 1 = 2 वर्ग इकाई
महत्वपूर्ण टिप्पणियाँ
ध्यान दें:
- समाकलन अवकलन की विपरीत प्रक्रिया है।
- अनिश्चित समाकलन में एक स्वेच्छ अचर C जुड़ता है।
- निश्चित समाकलन एक संख्यात्मक मान देता है।
- प्रतिस्थापन विधि में उचित प्रतिस्थापन का चयन महत्वपूर्ण है।
- खंडशः समाकलन में u और v का चयन ILATE नियम के अनुसार करें।
UP Board समाधान के लिए अधिक अभ्यास प्रश्नों का अभ्यास करें और समाकलन की विभिन्न विधियों में निपुणता प्राप्त करें।
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