If x 4x , dx = | 1 + x 1 - x | + | 1 + 2 x 1 - 2 x | + c , then the value of 32( + ^2) =

Step-by-step solution

Given integral is I = x 4x dx I = x 2 2x 2x dx = x 4 x x 2x dx I = 1 4 x 2x dx Multiplying the numerator and denominator by x : I = x 4 ^2 x 2x dx = x 4 (1 - ^2 x) (1 - 2 ^2 x) dx Substituting x = t x dx = dt : I = 1 4 1 (1 - t^2)(1 - 2t^2) dt Using partial fractions: 1 (1 - t^2)(1 - 2t^2) = 2 1 - 2t^2 - 1 1 - t^2 I = 1 4 ( 2 1 - 2t^2 - 1 1 - t^2 ) dt I = 1 2 1 1 - ( 2 t)^2 dt - 1 4 1 1 - t^2 dt Using the standard integral 1 a^2 - x^2 dx = 1 2a | a + x a - x | : I = 1 2 ( 1 2 2 | 1 + 2 t 1 - 2 t | ) - 1 4 ( 1 2 | 1