What is the integral of (sin(x))^5(cos(x))^3 ?

The integral of (sin(x))^5(cos(x))^3 is -(cos^4(x))/4+(cos^6(x))/3-(cos^8(x))/8+C

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integral of (sin(x))^5(cos(x))^3

\int (sin (x))^{5} (cos (x))^{3} d x

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

Solution steps

\int (sin (x))^{5} (cos (x))^{3} d x

Simplify

= \int sin^{5} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{3} (x) d x

Apply u-substitution

: \int - u^{3} (1 - u^{2})^{2} d u

= \int - u^{3} (1 - u^{2})^{2} d u

Expand

- u^{3} (1 - u^{2})^{2} : - u^{3} + 2 u^{5} - u^{7}

= \int - u^{3} + 2 u^{5} - u^{7} 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^{3} d u + \int 2 u^{5} d u - \int u^{7} d u

\int u^{3} d u = \frac{u ^{4}}{4}

\int 2 u^{5} d u = \frac{u ^{6}}{3}

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

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{3} - \frac{u ^{8}}{8}

Substitute back

u = cos (x)

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8}

Add a constant to the solution

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

\frac{d ^{2}}{d x ^{2}} (1 + \frac{1}{x ^{4}})

\frac{3}{x}

(\frac{d x}{d y}) + 3 x y^{3} = 0

x + 19

t^{\frac{1}{3}}

What is the integral of (sin(x))^5(cos(x))^3 ?
The integral of (sin(x))^5(cos(x))^3 is -(cos^4(x))/4+(cos^6(x))/3-(cos^8(x))/8+C