What is the integral of sin^5(x)*cos^5(x) ?

The integral of sin^5(x)*cos^5(x) is -(cos^6(x))/6+(cos^8(x))/4-(cos^{10}(x))/(10)+C

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integral of sin^5(x)*cos^5(x)

\int sin^{5} (x) \cdot cos^{5} (x) d x

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

Solution steps

\int sin^{5} (x) cos^{5} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{5} (x) d x

Apply u-substitution

: \int - u^{5} (1 - u^{2})^{2} d u

= \int - u^{5} (1 - u^{2})^{2} d u

Expand

- u^{5} (1 - u^{2})^{2} : - u^{5} + 2 u^{7} - u^{9}

= \int - u^{5} + 2 u^{7} - u^{9} 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 u^{5} d u + \int 2 u^{7} d u - \int u^{9} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int 2 u^{7} d u = \frac{u ^{8}}{4}

\int u^{9} d u = \frac{u ^{10}}{10}

= - \frac{u ^{6}}{6} + \frac{u ^{8}}{4} - \frac{u ^{10}}{10}

Substitute back

u = cos (x)

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10}

Add a constant to the solution

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

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\frac{d x}{d y} + \frac{1}{y ^{2}} x = \frac{1}{y ^{2}}

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What is the integral of sin^5(x)*cos^5(x) ?
The integral of sin^5(x)*cos^5(x) is -(cos^6(x))/6+(cos^8(x))/4-(cos^{10}(x))/(10)+C