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Question

# ∫2x(x2+1)(x2+2)dx is equal to

A
log(x2+1x2+2+C
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B
log(x21x2+2+C
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C
log(x2+1x22+C
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Solution

## The correct option is A log(x2+1x2+2∣∣+CHere, 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+2⇒1=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+1−1t+2)dt⇒I=ln(t+1|−ln(t+2|+c⇒I=ln(t+1t+2∣∣+cSubstuting back t=x2, we get⇒I=ln(x2+1x2+2∣∣+c

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