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The air pressure in the cabin of a passenger plane is modelled by the equation: P(x) = 3cos(x/2) - sin(x/2) where x is the altitude. Express P(x) in the form Rcos(x/2 +z) where z is acute and in degrees and then find the maximum pressure

This is a classic example of a trigonometry problem using the angle addition formulae. Recall that cos(A+B) = cos(A)cos(B) - sin(A)sin(B). Express R(x/2 -z) in similar form: Rcos(x/2 +z) = Rcos(x/2)cos(z) - ...
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Answered by Caspar P. • Maths tutor
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Integrate e^x sinx

Since we have two functions of x being multipied togrther, we have to integrate this by parts. Therefore if we say, u=sinx and v'=e x , then u'=cosx and v=e x .Applying the integration by parts rule of: uv' ...
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Answered by Kishan P. • Maths tutor
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Use integration by parts to integrate the following function: x.sin(7x) dx

Integration by parts follows the general form: ∫u (dv/dx) = u.v - ∫v (du/dx)Let x = uLet sin 7x = (dv/dx)∫x.sin(7x) dx = x.(-1/7)cos(7x) - ∫(-1/7)cos(7x).1 dx = (-x/7)cos(7x) + (1/49)sin(7x) + c
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Answered by Ahanna N. • Maths tutor
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Integrate 3x^2

3x^2 Integrate you add one to the power and divide by that number. In this case:3x^2 = (3x^3)/3 = x^3
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Answered by Jiano P. • Maths tutor
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The function f has domain (-∞, 0) and is defines as f(x) = (x^2 + 2)/(x^2 + 5) (here ^ is used to represent a power). Show that f'(x) < 0. What is the range of f?

First notice that f(x) = u/v. So f'(x) =[ v(u') - u(v')]/v 2 (the Quotient rule). After working it out, we find f'(x) = 6x/(x 2 + 5) 2 (the steps can be shown on the whiteboard). Since the denominator is alw...
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Answered by Chris C. • Maths tutor
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