Expert: Sherman D. - 4/10/2008

Question
Hi, I was wondering if you could help me with a couple of
calculus problems. If you could help me out and give me how
you did it step-by-step, I would appreciate it very much!!!

Thanks,
Morgan Ellyson
CMU

Solve:

log(7x^2+30x+2)-2log(x+1)=1

^=to the power of

also,..

sin(sin^-1(5/13)+cos^-1(3/5)

also,...

prove the identity:

(cosx+sinx)(1-sinxcosx)=cos^3x+sin^3x



THANKS FOR THE HELP!!!  

Answer
log(7x^2 + 30x + 2) - 2log(x + 1) = 1
log(7x^2 + 30x + 2) - log((x + 1)^2) = 1
log((7x^2 + 30x + 2)/((x + 1)^2)) = 1
(7x^2 + 30x + 2)/((x + 1)^2) = 10
7x^2 + 30x + 2 = 10(x + 1)^2
7x^2 + 30x + 2 = 10(x^2 + 2x + 1)
7x^2 + 30x + 2 = 10x^2 + 20x + 10
-3x^2 + 10x - 8 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-10 ± sqrt(10^2 - 4(-3)(-8)))/(2(-3))
x = (-10 ± sqrt(100 - 96))/(-6)
x = (-10 ± sqrt(4))/(-6)
x = (-10 ± 2)/(-6)
x = (-12/-6) or (-8/-6)
x = 2 or (4/3)

ANS = 2 or 1 1/3

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sin(sin^-1(5/13)) + cos^-1(3/5)

the sin and sin^-1 cancel out

assuming you wrote it correctly.

ANS : 53.515

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(cos(x) + sin(x))(1 - sin(x)cos(x)) = cos(x)^3 + sin(x)^3

using sum of cubes

cos(x)^3 + sin(x)^3 = (cos(x) + sin(x))(cos(x)^2 - sin(x)cos(x) + sin(x)^2)

sin(x)^2 + cos(x)^2 = 1, so

cos(x)^3 + sin(x)^3 = (cos(x) + sin(x))(1 - sin(x)cos(x))