यदि 4+x2x6dx A(4+x2)3/2(Bx26)x5+C है, तो A है:

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  1. -120
  2. 1120
  3. 120
  4. 1120

Answer (Detailed Solution Below)

Option 4 : 1120
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दिया गया है:

4+x2x6dx = A(4+x2)3/2(Bx26)x5+C

अवधारणा:

प्रतिस्थापन का उपयोग करके समाकलन की अवधारणा का प्रयोग करें..

गणना:

4+x2x6dx

फिर x=1tdx=1t2dt प्रतिस्थापित करने पर

=4+1t21t6(1t2)dt

=t34t2+1 dt

=t2t4t2+1 dt

गुणन नियम का उपयोग करके समाकलन करने पर

=[t2t4t2+1 dt(ddt(t2)t4t2+1 dt) dt]

फिर 4t2 + 1 = u ⇒ t dt = 1/8 du प्रतिस्थापित करने पर

=[t218u du(ddt(t2)18u du) dt]

=[t218u3/23/2(2t18u3/23/2) dt]

u का प्रतिस्थापन पूर्ववत करने पर

=[t218(4t2+1)3/23/2(2t18(4t2+1)3/23/2) dt]

=[112t2(4t2+1)3/216t(4t2+1)3/2dt]

पुनः 4t2 + 1 = u ⇒ t dt = 1/8 du प्रतिस्थापित करने पर

=[112t2(4t2+1)3/21618(u)3/2du]

=[112t2(4t2+1)3/2148(u)5/25/2]+C

u का प्रतिस्थापन पूर्ववत करने पर

=[112t2(4t2+1)3/21120(4t2+1)5/2]+C

t का प्रतिस्थापन पूर्ववत करने पर

=[1121x2(4x2+1)3/21120(4x2+1)5/2]+C

=[112(4+x2)3/2x51120(4+x2)5/2x5]+C

=1120(4+x2)3/2(x26)x5+C

इसलिए A = 1120 

अतः विकल्प (4) सही है।

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