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Given the equation x^3-12x^2+ax-48=0 has roots p, 2p and 3p, find p and a.

QUESTION: Given the equation x 3 -12x 2 +ax-48 = 0 has roots p, 2p and 3p, find p and a. Roots mean x = p, x = 2p and x = 3p hence (x-p), (x-2p) and (x-3p) are factors of the equation. Expanding these three ...
MD
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a) Show that d/dx(arcsin x) = 1/(√ (1-x²)). b) Hence, use a suitable trigonometric substitution to find ∫ (1/(√ (4-2x-x²))) dx.

Will be easier to explain with whiteboard!a) Let y = arcsin x. sin y = x cos y dy/dx = 1 dy/dx = 1/(cos y) dy/dx = 1/(√ (1 - sin 2 y)) dy/dx = 1/(√( 1 - x 2 )) as required.b) 4 - 2x - x 2 = - (x 2 + 2x - 4) ...
ED
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write the sum cos(x)+cos(2x)+...+cos(nx) as a quotient only involving sine and cosine functions

We can write this sum S as Re(e^ix+e^2ix+...+e^nix), we now have a finite geometric series, which we know the formula for.Have, S = Re( e^ix(1-e^inx)/(1-e^ix)) - Now factoring numerator and denominator to lo...
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How do i figure out if integrals are improper or not and how do i know which limit is undefined?

Improper integrals can take many forms. The first step when dealing with a potentially improper integral is to test the limits of the integral i.e. a and b. If these are at infinity/negative infinity, or the...
HD
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Let I(n) = integral from 1 to e of (ln(x)^n)/(x^2) dx where n is a natural number. Firstly find I(0). Show that I(n) = -(1/e) + n*I(n-1). Using this formula find I(1).

For the first part this is a standard polynomial integral whereby we can follow the power rule for anti-differentiation. In part ii), if we look at the question we see that we're trying to get I(n-1) in our ...
MP
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