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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

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

= 5 \cdot \int - 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

= 5 (- \int u^{5} d u + \int u^{7} d u)

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

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

= 5 (- \frac{u ^{6}}{6} + \frac{u ^{8}}{8})

Substitute back

u = cos (x)

= 5 (- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8})

Add a constant to the solution

= 5 (- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8}) + C

\int x^{2} + 4 x - 5 d x

\frac{\partial}{\partial x} (- 3 cos (π x) + 6 x^{2} y + 5 e^{x z} + 3 yz - 98)

\int x - \frac{8}{x} d x

ln^{2} (1 + x)

n = 1 \sum \infty \frac{9}{π ^{n}}

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