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Question

# ∫(2x+3)√x2+3x+2dx is equal to.

A

32(x2+3x+2)3/2+ C
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B

32(x2+3x+2)2/3 + C
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C

23(x2+3x+2)2/3 + C
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D

23(x2+3x+2)3/2 + C
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Solution

## The correct option is D 23(x2+3x+2)3/2 + CTo solve these types of integrals we express the linear term as a sum of derivative of the quadratic term and a constant term. Suppose the integral to be I=∫(ax+b)√cx2+dx+edx Here, (ax+b)=Addx(cx2+dx+e)+B For this case, (2x+3)=Addx(x2+3x+2)+B ⇒(2x+3)=A(2x+3)+B Now, comparing the coefficients of x and the constant term we get A=1 and B=0. Now we can write our integral as: I=∫(2x+3)√x2+3x+2dx ⇒I=∫(ddx(x2+3x+2)} ×√x2+3x+2dx Now we substitute t=(x2+3x+2) then, dt=(2x+3)dx substituting these back in the integral we get, I=∫t1/2dt which gives I=t3/23/2=23t3/2+C substituting back t we get I=23(x2+3x+2)3/2+C

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