Question

\( \int_{3 / 2}^{9 / 2} \sqrt{27+12 x-4 x^{2}} d x \)

Answer 1

​​​​​​​​​​​​​​​​​​wkt   (27+12x-4x²)^1/2 = (6²-(2x-3)²)^1/2  ​​​​​​​   ​​​​​​​

wkt integral of (a²-x²)^1/2= x​​​​​​​​​​​​​​/2(a²-x²)^1/2 + a²/2 sin^-1(x/a)

therefore applying the same concept to above equation we get

intergal of (6²-(2x-3)²)^1/2 = (2x-3)/2 (6²-(2x-3)²)^1/2 + 6²/2 sin^-1(2x-3/6) _/ d/dx(2x-3)

by applying the limits from 3/2 to 9/2 we get the answer