What is the integral of sin^3(x)cos^2(y) ?

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

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integral of sin^3(x)cos^2(y)

\int sin^{3} (x) cos^{2} (y) d x

cos^{2} (y) (- cos (x) + \frac{cos ^{3} ( x )}{3}) + C

Solution steps

\int sin^{3} (x) cos^{2} (y) d x

Take the constant out:

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

= cos^{2} (y) \cdot \int sin^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

\int 1 d u = u

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

= cos^{2} (y) (- u + \frac{u ^{3}}{3})

Substitute back

u = cos (x)

= cos^{2} (y) (- cos (x) + \frac{cos ^{3} ( x )}{3})

Add a constant to the solution

= cos^{2} (y) (- cos (x) + \frac{cos ^{3} ( x )}{3}) + C

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