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Rewrite ln(x) as 1ln(x) then integrate by parts. The formula for integration by parts is ∫ uv' = uv - ∫ vu', here use u = ln(x) and v' = 1. By differentiating u we get u' = 1/x, an...
The key concepts to apply in this question will be the product and chain rules, namely: if f(x)=g(x)h(h), then f'(x)=g(x)h'(x) + g'(x)h(x), and if h(x)=u(v(x)), then h'(x)=u'(v(x))v'(x).
Equivalen...
We start by expanding out the double trigonometric terms (sin(2a)) using the double angle formula, giving us: [2sin(a)cos(a)] / [1+cos^2(a) - sin^2(a)]. Next we spot that on the denominator (bottom half o...
The definite integral gives the value of the area bounded between the curve at the given limits, and the x-axis (if we are integrating with respect to x!) If this value is negative, it means that the area...
using the product rule
product rule is dy/dx=v.du/dx+u.dv/dx
let u = x^2 and v = cos(x)
du/dx=2x
dv/dx = -sin(x)
so dy/dx=2xcos(x)-x^2sin(x)
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