simplifying using trig identities?

Question

I'm trying to integrate cos^(3)xsinx dx
I arrived at an answer of (1/2)sin^(2)x - (1/4)sin^(4)x + C
The answer in the back of the book is (-1/4)cos^(4)x + C
How do I get from A to B? I'm assuming there's a simple trig identity involved, but I can't figure that out. Thanks for the help!

Answer 1

Ignore C for now, since the constants for the two solutions are different. Factor out sin^2x. We get:

1/4 * sin^2(x) * (2+sin^2(x))

Let sin^2(x) = 1 - cos^2(x)
so that 2+sin^2(x) = 1+cos^2(x)

Then we get 1/4 * (1-cos^2(x)) * (1+cos^2(x)) =
1/4 ( 1 - cos^4(x)) = 1/4 - 1/4*cos^4(x)

Steve

Answer 2

to integrate simply make the substitution u=cos x , then du/dx = -sin x and
int. cos^3 x . sin x dx = - int. u^3 dx = -u^4/4 +c
since u= cos x, the answer is,

(-1/4)cos^4 x + c
as required.

Answer 3

oh man thats ... that is simple...........
∫cos^(3)xsinx dx
∫cos^(3)x (sinx) dx the integral of cos is (-sin)
(-1)∫[cosx]^(3) (-sinx )dx cos^(3)x means cos x whole cube

now just apply basic power addition and division law
{ -[cosx]^ (3+1) }/ (3+1) +c
-1/4 cos^(4) +c answer

Answer 4

so as that first area expands and also you get cosx + (cosx * tan^2x) improve out tan^2x cosx + (cosx * (sin^2x/cos^2x)) Multiply and simplify the words interior the parentheses cosx + (sin^2x/cosx) Get both words over the same time period only so as that they could be able to upload genuinely (positioned cosx over one, multiply the sin^2x through the denom of the cosx time period and multiply the cosx through the sin^2x denom and then multiply the denoms) (cos^2x/cosx) + (sin^2x/cosx) combine the words because they're less than the same fraction (cos^2x+sin^2x)/cosx cos^2x+sin^2x = a million in trigonometric identities so it reduces to at least a million/cosx Which also equals secx playstation that is strictly what the guy above me printed yet minus the a million+tan^2x shortcut. there are various of concepts to do those identities and he only did the better one, yet I had forgotten the shortcut and took the easy way. both are precise.