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Question

The integral x+2(x2+3x+3)x+1dx is equl to

A
13tan1[x3(x+1)]+C
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B
23cot1(3(x+1)x)+C
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C
23cot1[xx+1]+C
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D
13cot1[x3x+1]+C
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Solution

The correct option is B 23cot1(3(x+1)x)+C
Consider, I=(x+2)dx(x2+3x+3)x+1

Let x+1=t2dx=2tdt

Therefore, 2(t2+1)tdt[(t21)2+3t2].t=2(t2+1)dt[(t4+t2+1)=2(t2+1)dt[(t2t+1)(t2+t+1)

=(t2t+1)+(t2+t+1)dt[(t2t+1)(t2+t+1)=dtt2+t+1+dtt2t+1

=23[tan1(23(t+12))+tan1(23(t12))]=23tan1⎢ ⎢ ⎢ ⎢4t3143(t214)⎥ ⎥ ⎥ ⎥

=23tan1[t31t2]=23tan1(3(x+1)x)

Therefore, I=23cot1(3(x+1)x)+c

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