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Let's take for example the integral 5x²(x³-4)⁴dx. This is very difficult and not at all nice to expand in terms of x, so we can effectively "create" a new variab...

AB = B - A = (4,3,-6)- (2,0,-1) = (2 , 3 , -5)BC = C - B = (9,3,-4) - (4,3,-6) = (5, 0 , 2)
If perpendicular "AB dot BC" should equal zeroAB dot BC = (2,3,-5) dot (5,0,2) = (2x5) + (3x0) + ...

Ok so for this question we'll use this example: y = (e^{-2x^2} + 2)^1/5
We start off by making the expression inside the brackets equal to u. In other words u = e^{-2x^2} + ...

First make a substitution so we can apply the power rule ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C more simply. Can see u=x+2 means (x+2)^1/2 -> u^1/2 and so will help us appl...

differentiating: xy'+y-2x+(x/y)y'+log(y)=0rearranging: y'=y(2x-y-log(y))/x(1+y)at (2,1): y'=3/4 so gradient of normal at (2,1) is -4/3so the equation of the normal is y-1=(-4/3)(x-2)which is equivalent to...