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

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

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

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

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

Solution steps

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

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 t )}{3} + \frac{2 cos ^{5} ( 2 t )}{5} - \frac{cos ^{7} ( 2 t )}{7})

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{1}{8 x})

\int - t e^{t} d t

\int \frac{1}{10 - x ^{2}} d x

y^{^{'}} + 2 x y = x, y (0) = 3

\frac{d ^{2}}{d x ^{2}} ((cos (2 x)) (sin (6 x)))

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