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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...

Let f denote an integral sign, I will write the integrand in square brackets. The formula for integration by parts is given by:f [(u)(dv/dx)]dx = uv - f [((du/dx)(v)]dxTo apply ...

First we need to use trigonometric identities to convert the sin^3(x) to a single power. This is because we cannot integrate trigonometric functions that are above the power of 1. We need to use the doubl...

We know that f'(x) = x^3 + 2*x^0.5 + 8, and we can integrate both sides. This gives us f(x) = (1/4)*x^4 + (4/3)x^(3/2) + 8x + c, remembering to add the constant of integration. Now, we are almost...