What is the integral of cos^5(x) ?

The integral of cos^5(x) is sin(x)-(2sin^3(x))/3+(sin^5(x))/5+C

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integral of cos^5(x)

\int cos^{5} (x) d x

sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5} + C

Solution steps

\int cos^{5} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= \int 1 - 2 u^{2} + u^{4} 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 2 u^{2} d u + \int u^{4} d u

\int 1 d u = u

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

\int u^{4} d u = \frac{u ^{5}}{5}

= u - \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

Substitute back

u = sin (x)

= sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5}

Add a constant to the solution

= sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5} + C

x \to 9 lim (cos (\frac{π x}{4}))

y \to \infty lim (\frac{2 y ^{2} - 3 y + 5}{y ^{2} - 5 y + 2})

x \to 2 lim (x^{3} + 3 x^{2})

f (x) = 8 x^{2} - x^{3}

\int x^{3} x^{2} + 17 d x

What is the integral of cos^5(x) ?
The integral of cos^5(x) is sin(x)-(2sin^3(x))/3+(sin^5(x))/5+C