What is the integral of sin^5(2x)cos^2(2x) ?

The integral of sin^5(2x)cos^2(2x) is -1/6 cos^3(2x)+1/5 cos^5(2x)-1/14 cos^7(2x)+C

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integral of sin^5(2x)cos^2(2x)

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

- \frac{1}{6} cos^{3} (2 x) + \frac{1}{5} cos^{5} (2 x) - \frac{1}{14} cos^{7} (2 x) + C

Solution steps

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

Apply u-substitution

: \int sin^{5} (u) cos^{2} (u) \frac{1}{2} d u

= \int sin^{5} (u) cos^{2} (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sin^{5} (u) cos^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int (1 - cos^{2} (u))^{2} sin (u) cos^{2} (u) d u

Apply u-substitution

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

= \frac{1}{2} \cdot \int - v^{2} (1 - v^{2})^{2} d v

Expand

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

= \frac{1}{2} \cdot \int - v^{2} + 2 v^{4} - v^{6} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int v^{2} d v + \int 2 v^{4} d v - \int v^{6} d v)

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

\int 2 v^{4} d v = \frac{2 v ^{5}}{5}

\int v^{6} d v = \frac{v ^{7}}{7}

= \frac{1}{2} (- \frac{v ^{3}}{3} + \frac{2 v ^{5}}{5} - \frac{v ^{7}}{7})

Substitute back

= \frac{1}{2} (- \frac{cos ^{3} ( 2 x )}{3} + \frac{2 cos ^{5} ( 2 x )}{5} - \frac{cos ^{7} ( 2 x )}{7})

Simplify

\frac{1}{2} (- \frac{cos ^{3} ( 2 x )}{3} + \frac{2 cos ^{5} ( 2 x )}{5} - \frac{cos ^{7} ( 2 x )}{7}) : - \frac{1}{6} cos^{3} (2 x) + \frac{1}{5} cos^{5} (2 x) - \frac{1}{14} cos^{7} (2 x)

= - \frac{1}{6} cos^{3} (2 x) + \frac{1}{5} cos^{5} (2 x) - \frac{1}{14} cos^{7} (2 x)

Add a constant to the solution

= - \frac{1}{6} cos^{3} (2 x) + \frac{1}{5} cos^{5} (2 x) - \frac{1}{14} cos^{7} (2 x) + C

f (x) = 2 x^{2} + 7

\frac{d y}{d x}, y = 3 x^{5}

\frac{\partial}{\partial y} (\frac{x}{x ^{4} - y ^{7}})

f (x) = \frac{x ^{3}}{8}

\int \frac{ln ( 5 x )}{x} d x

What is the integral of sin^5(2x)cos^2(2x) ?
The integral of sin^5(2x)cos^2(2x) is -1/6 cos^3(2x)+1/5 cos^5(2x)-1/14 cos^7(2x)+C