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integral of (tan(x))dx using the substitution u = cos(x)

given u = cos(x), therefore du/dx=-sin(x), as tan(x)=sin(x)/cos(x), can rewrite tan(x)=(-du/dx)/u, therefore integral can become [(-1/u)du], after inegrating you are left with -ln(u)+c, therefore ln(1/u)+c, ...
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Answered by Frederick R. • Maths tutor
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f(x) = x^x, find f'(3).

Therefore, y = x x can then natural log both sides leaving ln(y) = xln(x) then differentiating both sides wrst to x d/dx(ln(y)=xln(x))we are then left with this expression (dy/dx)(1/y)=ln(x)+1 multiplying up...
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Answered by Frederick R. • Maths tutor
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A 10 kilogram block slides down a 30 degree inclined slope, the slope has a coefficient of friction of 0.2. Calculcate the blocks acceleration down the slope.

Resolving all forces in the vertical gives the normal force as 10gcos(30). Resolving all forces normal to the slope gives the frictional force as 2gcos(30) (given Friction = mu*R). Using second law, F=ma. 10...
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Answered by Frederick R. • Maths tutor
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How do I maximise/minimise a given function f(x)?

To find an extreme point of a function you must first take the derivative of f(x) with respect to x. As the function will peak/trough at the extreme point, the gradient at this point will be equal to 0 and t...
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The curve C has the equation y=3x/(9+x^2 ) (a) Find the turning points of the curve C (b) Using the fact that (d^2 y)/(dx^2 )=(6x(x^2-27))/(x^2+9)^3 or otherwise, classify the nature of each turning point of C

(a)To find the turning points of a curve, need to solve dy/dx=0. Using the quotient rule one can differentiate y:y=f(x)/g(x) dy/dx=(f'(x)g(x)-f(x)g'(x))/(g(x)) 2 f(x)=3x, f'(x)=3, g(x)=(9+x 2 ), g'(x)=2x→ dy...
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Answered by Liora C. • Maths tutor
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