What is the integral of (cos^3(x)) ?

The integral of (cos^3(x)) is sin(x)-(sin^3(x))/3+C

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integral of (cos^3(x))

\int (cos^{3} (x)) d x

sin (x) - \frac{sin ^{3} ( x )}{3} + C

Solution steps

\int cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) d x

Apply u-substitution

: \int 1 - u^{2} d u

= \int 1 - u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= u - \frac{u ^{3}}{3}

Substitute back

u = sin (x)

= sin (x) - \frac{sin ^{3} ( x )}{3}

Add a constant to the solution

= sin (x) - \frac{sin ^{3} ( x )}{3} + C

x \to 3 lim (5 x + 2)

x \to 4 lim (32 - x^{2})

\frac{d y}{d x} = \frac{2 x y}{1 + x ^{2}}

g (x) = 7 x, at x = 9

\frac{d y}{d t} = 1600 - \frac{4 y}{170}

What is the integral of (cos^3(x)) ?
The integral of (cos^3(x)) is sin(x)-(sin^3(x))/3+C