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this is an example of an integration by parts problem, we must use integration by parts to evaluate this integral;although this would not be entirely obvious as the integral does not seem to be the produc...

(2x)/2= 2x^-2d2y/dx2= -4x^-1 - 1/4

So we have to integrate these 3 terms one by one. To integrate 2x⁵ you add 1 to the power of x, so that becomes 6, you then divide the coefficient by the new power of x. So 3/6=1/2 giving the i...

differentiate both side with respect to x : (dy/dx)e^(-2x)+y(-2e^(-2x)) = 2+2y(dy/dx)
rearrange it : (-2y+e^(-2x))(dy/dy) = 2 + 2ye^(-2x) ==> dy/dx = ( 2 + 2ye^(-2x) ) / ( -2y+e^(-2x) )

Let f denote an integral sign, I will write the integrand in square brackets. We can use integration by parts to integrate ln(x), the "trick" here is to imagin ln(x) as 1 x ln(x). Integ...