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Question

2x(x2+1)(x2+2)dx is equal to

A
logx2+1x2+2+C
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B
logx21x2+2+C
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C
logx2+1x22+C
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Solution

The correct option is A logx2+1x2+2+C
Here, if we make a substitution t=x2,we get dt=2xdx.
Now, our integral becomes:
dt(t+1)(t+2).
Now, the integrand is a proper fraction, so we can use integration by partial fraction.
Now, we can write the integrand as:
1(t+1)(t+2)=at+1+bt+21=a(t+2)+b(t+1)comparing coefficient of x and constant terms, we get:a+b=0 and 2a+b=1solving these two questions, we get:a=1 and b=1.So, our integral becomes:I=(1t+11t+2)dtI=ln(t+1|ln(t+2|+cI=ln(t+1t+2+cSubstuting back t=x2, we getI=ln(x2+1x2+2+c

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