help please...on questions that confuse me.. (1 Viewer)

[{*(00)*}]

^ = to the power of, by the way.
∏ = pi

(sorry it's a little difficult using the keyboard)

1. if f(x)= 2 cos ^-1(x / √2) - sin^-1(1 - x^2)

a)find f '(x)

b)hence,or otherwise, show that
2 cos ^-1(x / √2) - sin^-1(1 - x^2)= ∏ / 2



2. by considering sin^3 = sinx (1- (cos^2)x)
find

a) ∫ tan 3x dx
b) ∫ sin 3x dx


thanks! mike.

 

[Mountain.Dew]

^ = to the power of, by the way.
∏ = pi

(sorry it's a little difficult using the keyboard)

1. if f(x)= 2 cos ^-1(x / √2) - sin^-1(1 - x^2)

a)find f '(x)

b)hence,or otherwise, show that
2 cos ^-1(x / √2) - sin^-1(1 - x^2)= ∏ / 2



2. by considering sin^3 = sinx (1- (cos^2)x)
find

a) ∫ tan 3x dx
b) ∫ sin 3x dx


thanks! mike.

1) (a) apply the chain rule to this question, so u have to differentiate INSIDE the trig inverse function, as a consequence of this chain rule.
(b) simply integrate the function, then sub in x = 0, y = (something) to get ur c, ur constant.

2) a) for this, try to find an expression for tanx in terms of sin's and cos's OTHER than tanx = sinx / cosx. then, simply substitute x = 3x to get tan3x = (something in terms of coses and sins), then, to integrate, make some wise decisions when doing some clever and handy substitutions...
(b) i would simply go to -1/3cos(3x) + c, but we have to consider sin^3 x= sinx (1- (cos^2)x)

i would try to find an expression for sin3x in terms of sinx's, then apply that to the required equation. then, like in (a), do some clever substitutions to get something integrable, then get ur answer.

 

[icycloud]

For question 2, I think he meant ∫tan³xdx and ∫sin³xdx

Assuming this:

∫tan³xdx = ∫sin³x/cos³xdx
= ∫sinx(1-cos²x)/cos³xdx
= ∫sinx/cos³xdx - ∫sinxcos²x/cos³xdx
= ∫-d(cosx)/(cosx)³ - ∫sinx/cosxdx
= sec²x/2 + ln(cosx) + c

∫sin³xdx = ∫sinx(1-cos²x)dx
= ∫sinxdx - ∫sinxcos²xdx
= -cosx - ∫-d(cosx)(cosx)²
= -cosx + cos³x/3 + c

Alternate method for ∫tan³xdx without using the previous part:

∫tan³xdx = ∫tanx(sec²+1)dx
= ∫tanxsec²xdx + ∫tanxdx
= ∫secxd(secx) + ln(cosx)
= sec²x/2 + ln(cosx) + c (as before)