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find : ∫(3x+1)√(4-3x-2x^2)dx - CBSE (Science) Class 12 - Mathematics

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Question

 

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

 

Solution

 
 

`Let 3x+1=λd/dx(4−3x−2x^2)+μ`

3x+1=λ(34x)+μ

3x+1=3λ+μ4λx

3=4λ , 3λ+μ=1

λ=3/4, μ=5/4

`I=int(3x+1)sqrt(4-3x-2x^2)dx`

`=∫[−3/4(−3−4x)−5/4]sqrt(4−3x−2x^2)dx`

`=∫−3/4(−3−4x)sqrt(4−3x−2x^2)dx−∫5/4 sqrt(4−3x−2x^2)dx`

`=−3/4∫(−3−4x)sqrt(4−3x−2x^2)dx−5/4∫sqrt(4−3x−2x^2)dx  `

Let 43x2x2=t in the first integral(34x)dx=dt 

`∴ I=−3/4∫sqrtt dt−5/4∫sqrt(−2(x^2+3/2x−2)dx`

`=−3/4×2/3t^(3/2)+C_1−5/4∫sqrt(−2(x^2+3/2x−2+9/16−9/16)dx`

`=−1/2(4−3x−2x^2)^(3/2)+C_1−5/4∫sqrt(−2[(x+3/4)^2−(sqrt41/4)^2])dx`

`=−1/2(4−3x−2x^2)^(3/2)+C_1−(5sqrt2)/4∫sqrt((sqrt41/4)^2−(x+3/4)^2)dx`

`=−1/2(4−3x−2x^2)^(3/2)+C_1-(5sqrt2)/4[1/2(x+3/4)sqrt((41/16)−(x+3/4)^2)+1/2(41/16)sin^−1 ((x+3/4)/(sqrt41/4))+C_2]`


`=−1/2(4−3x−2x^2)^(3/2)−5/(4sqrt2)(x+3/4)sqrt((41/16)−(x+3/4)^2)-205/(64sqrt2) sin^−1 ((4x+3)/sqrt41)+C, `

where C=C_1C_2

 
 
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Solution find : ∫(3x+1)√(4-3x-2x^2)dx Concept: Integrals of Some Particular Functions.
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