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To integrate ln(x) we will have to use integration by parts. 

The equation for integration by parts is: (then equation written on the whiteboard).

In the case of ln(x), let v=ln(x) ...

Volume=81 cm³ 

b=81/2x² 

L=4x+4(2x)+ 4(b)

L=4(x)+4(2x)+4(81/2x²) 

L=12x+ 162/x²

You would use the product rule. uv'=uv- u'vdx. In this case we would allocate u= ln x and v'=1 so u'=1/x and v=x so uv=xlnx whilst u'v=x/x=1 so we would have xln(x) -1dx. Next we would get xln(x)-x +c

Option 1 - Differentiate using product rule giving dy/dx = cos²(x) - sin²(x). (2 marks) Subbing in x as pi (1 mark) then gives (-1)² + (0)² . ...

let us start by taking two points on the curve y=x², the first with x-coordinate x, and the second x+Δx and drawing a straight line between them. We can form a right-angled triangle, with two s...