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

ConceptIntegrals of Some Particular Functions

#### 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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