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

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

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integral of cos^5(t)

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

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

Solution steps

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

Rewrite using trig identities

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

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 (t)

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

Add a constant to the solution

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

cos (π - 6 x)

y = 2 sin (x) + c e^{- x} cos (x)

18 y^{^{''}} + 9 y^{^{'}} - 2 y = 0

y^{^{'}} = 3 x^{2} + 1

\frac{5 s - 2}{7 s ^{2} - s - 11}

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