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

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

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integral of cos^5(y)

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

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

Solution steps

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

Rewrite using trig identities

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

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 (y)

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

Add a constant to the solution

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

\frac{d}{d x} (3 + 8 x)

\int \frac{e ^{x}}{( e ^{x} - 2 ) ( e ^{2 x} + 1 )} d x

\frac{\partial}{\partial x} ((e)^{x + 2 y})

x^{\frac{9}{6}}, 5

\frac{d}{d x} (10500 x + \frac{1}{5} x^{5} + 1900)

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