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

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

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integral of cos^3(y)

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

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

Solution steps

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

Rewrite using trig identities

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

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 (y)

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

Add a constant to the solution

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

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

y^{^{'}} - 3 y = e^{2 t}

1 - (1 - x)^{2 - 1}

ln (1 - 4 x)

\int e^{t^{2} - t} d t

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