What is the integral of 5cos^5(x)sin^4(x) ?

The integral of 5cos^5(x)sin^4(x) is sin^5(x)-10/7 sin^7(x)+5/9 sin^9(x)+C

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integral of 5cos^5(x)sin^4(x)

\int 5 cos^{5} (x) sin^{4} (x) d x

sin^{5} (x) - \frac{10}{7} sin^{7} (x) + \frac{5}{9} sin^{9} (x) + C

Solution steps

\int 5 cos^{5} (x) sin^{4} (x) d x

Take the constant out:

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

= 5 \cdot \int cos^{5} (x) sin^{4} (x) d x

Rewrite using trig identities

= 5 \cdot \int (1 - sin^{2} (x))^{2} cos (x) sin^{4} (x) d x

Apply u-substitution

: \int u^{4} (1 - u^{2})^{2} d u

= 5 \cdot \int u^{4} (1 - u^{2})^{2} d u

Expand

u^{4} (1 - u^{2})^{2} : u^{4} - 2 u^{6} + u^{8}

= 5 \cdot \int u^{4} - 2 u^{6} + u^{8} 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

= 5 (\int u^{4} d u - \int 2 u^{6} d u + \int u^{8} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 2 u^{6} d u = \frac{2 u ^{7}}{7}

\int u^{8} d u = \frac{u ^{9}}{9}

= 5 (\frac{u ^{5}}{5} - \frac{2 u ^{7}}{7} + \frac{u ^{9}}{9})

Substitute back

u = sin (x)

= 5 (\frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9})

Simplify

5 (\frac{sin ^{5} ( x )}{5} - \frac{2 sin ^{7} ( x )}{7} + \frac{sin ^{9} ( x )}{9}) : sin^{5} (x) - \frac{10}{7} sin^{7} (x) + \frac{5}{9} sin^{9} (x)

= sin^{5} (x) - \frac{10}{7} sin^{7} (x) + \frac{5}{9} sin^{9} (x)

Add a constant to the solution

= sin^{5} (x) - \frac{10}{7} sin^{7} (x) + \frac{5}{9} sin^{9} (x) + C

y = x^{3} - 2 x^{2} + 5, at x = 2

ln (π)

x \to - 0.1 lim (\frac{sin ( x )}{x})

n = 1 \sum \infty \frac{1}{7 + e ^{- n}}

n \to \infty lim (3^{n} 7^{- n})

What is the integral of 5cos^5(x)sin^4(x) ?
The integral of 5cos^5(x)sin^4(x) is sin^5(x)-10/7 sin^7(x)+5/9 sin^9(x)+C