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

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

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integral of cos(x)sin^3(y)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= cos (x) \cdot \int sin^{3} (y) d y

Rewrite using trig identities

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

Apply u-substitution

: \int - 1 + u^{2} d u

= cos (x) \cdot \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

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

\int 1 d u = u

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

= cos (x) (- u + \frac{u ^{3}}{3})

Substitute back

u = cos (y)

= cos (x) (- cos (y) + \frac{cos ^{3} ( y )}{3})

Add a constant to the solution

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

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