Evaluate: ∫3+4x-4x2 dx - Mathematics and Statistics

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Sum

Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`

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Solution

Let I = `intsqrt(3 + 4x - 4x^2)  dx`

= `2int sqrt(3/4 + x - x^2)  dx`

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

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

= `2int sqrt(1 - (x - 1/2)^2)  dx`

= `2[(x - 1/2)/2 sqrt(1 - (x - 1/2)^2) + 1/2 sin^-1(x - 1/2)] + c`

= `2[(2x - 1)/4 sqrt(3/4 + x - x^2) + 1/2 sin^-1((2x - 1)/2)] + c`

∴ I = `(2x - 1)/4 sqrt(3 + 4x - 4x^2) + sin^-1((2x - 1)/2) + c`

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