What is the integral of cos cubed x?

Answer 1

cos^3(x) = cos^2(x) cos(x)

= (1-sin^2(x))·cos(x)

=cos(x) - sin^2(x)·cos(x)

Integrating, the integral of cos(x) is sin(x) -----(1)

To integrate sin^2(x) cos(x)·dx, let sin(x) =u

cos(x)·dx = du

The integral becomes integral (u^2)·du = u^3/3 = (sin^3(x))/3 --(2)

(1)-(2) gives the answer to your question

sin(x) - (sin3(x))/3 +C

*Edit*

When we are solving the integral of cos(x) - sin^2(x)·cos(x),

& the two terms are split up, do not forget to take our signs into account, in this case;

the positive or negative constants/coefficients within what we are taking the integral of.

So here, ∫(cos(x) - sin^2(x)·cos(x))

is split up into ∫cos(x) minus ∫sin^2(x)·cos(x)

the 2nd integrand (function which is being integrated)

-sin^2(x)·cos(x)

is the same as

(-1)·sin^2(x)·cos(x)

So when taking the integral of this integrand, pull -1, a constant, out & place it in front of the integral symbol , ∫.

So once we substitute u for sin(x) & integrate our integrand (original function) which we have now put into terms of u & du, and get (u^3)/3, as the integral of sin^2(x)·cos(x)) [again, (u^3)/3 is in terms of du, when u=sin(x)] we simply cannot ignore the fact that we are subtracting that integral from that of cos(x), no matter how easy it seems to just ignore our negatives.

W.e are subtracting ∫(u^2)·du from whatever the ∫cos(x) is equal to, not adding them.

sin(X) MINUS (sin^3(x))/3 + C

should be the correct answer.

ANOTHER WAY CAN BE SUBSTITUTING THE VALUE OF COS^3X FROM COS3X i.e

INTEGRATE COS^3X....................................1

4COS^3X- 3COSX

4COS^3X- 3COSX= COS3X

COS^3X=(COS3X+3COX)/3

SUBSTITUTING THE VALUE OF COS^3X IN EQUATION 1

INTEGRAL (COS3X+3COSX)/3

AS 1/3 IS CONSTANT IT CAME OUT OF INTEGRAL

INTEGRAL OF COS3X=SINX/3

INTEGRAL OF 3COSX= 3SINX..........................................................