What is the integral of cos^7(y) ?

The integral of cos^7(y) is sin(y)-sin^3(y)+(3sin^5(y))/5-(sin^7(y))/7+C

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integral of cos^7(y)

\int cos^{7} (y) d y

sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7} + C

Solution steps

\int cos^{7} (y) d y

Rewrite using trig identities

= \int (1 - sin^{2} (y))^{3} cos (y) d y

Apply u-substitution

: \int (1 - u^{2})^{3} d u

= \int (1 - u^{2})^{3} d u

Expand

(1 - u^{2})^{3} : 1 - 3 u^{2} + 3 u^{4} - u^{6}

= \int 1 - 3 u^{2} + 3 u^{4} - u^{6} 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 3 u^{2} d u + \int 3 u^{4} d u - \int u^{6} d u

\int 1 d u = u

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

\int 3 u^{4} d u = \frac{3 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= u - u^{3} + \frac{3 u ^{5}}{5} - \frac{u ^{7}}{7}

Substitute back

u = sin (y)

= sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7}

Add a constant to the solution

= sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7} + C

\frac{d}{d x} (- 5 cot (x))

\int \frac{y}{( y + 2 ) ( 3 y - 1 )} d y

\frac{d x}{d t} - 0.06 x = 200

2 + \frac{8}{x} + \frac{6}{x ^{2}}

(- sin (x))^{^{'}}

What is the integral of cos^7(y) ?
The integral of cos^7(y) is sin(y)-sin^3(y)+(3sin^5(y))/5-(sin^7(y))/7+C