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How do I integrate (sin x)^6?

Letting z be the complex number cos x + i sin x, we will use the identities (z n + 1/z n ) = 2 cos nx and (z n - 1/z n ) = 2i sin nx. Both of these follow from De Moivre’s theorem and the odd and even proper...
KD
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Find the GS to the following 2nd ODE: d^2y/dx^2 + 3(dy/dx) + 2 = 0

Set up the auxiliary equation by letting (dy/dx) = m So we have: m 2 + 3m + 2 = 0 Solve for m and we get: (m+1)(m+2) = 0Therefore, m 1 =-1 and m 2 =-2 Now we see we have 2 different real numbers as the solut...
IG
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Prove De Moivre's by induction for the positive integers

First we check the formula for our first case, n=1 where (cosx + isinx) n =cosnx + isinnx (cosx + isinx) 1 = cos(1x) + isin(1x) - holds true for n=1 next we assume true for our case n=k(cosx + isinx) k = cos...
HS
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Given that k is a real number and that A = ((1+k k)(k 1-k)) find the exact values of k for which A is a singular matrix.

okay so A is a 2x2 matrix.for it to be singular its determinate has to equal 0.a 2x2 matrix's determinate is equal to m 1,1 m 2,2 - m 1,2 m 2,1 for this example:det(A) = (1+k)(1-k) - (k)(k) = 0multiplying ou...
KY
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A rectangular hyperbola has parametric equations x = 4t, y = 4/t , (z non 0). Points P and Q on this hyperbola have parameters t = 1/4 and t = 2. Find the equation of the line l which passes through the origin and is perpendicular to the line PQ.

Since we are told that the line l is perpendicular to the line PQ, we first need to find out the gradient of PQ. To do this, it is easiest to find out the co-ordinates of points P and Q in terms of their car...
ZE
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