What is the integral of 2sin^2(xco)s^3x ?

The integral of 2sin^2(xco)s^3x is (s^3)/(4c^2o^2)(2c^2o^2x^2-2coxsin(2cox)-cos(2cox))+C

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integral of 2sin^2(xco)s^3x

\int 2 sin^{2} (x co) s^{3} x d x

\frac{s ^{3}}{4 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x)) + C

Solution steps

\int 2 sin^{2} (x co) s^{3} x d x

Take the constant out:

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

= 2 s^{3} \cdot \int sin^{2} (co x) x d x

Apply u-substitution

: \int \frac{u sin ^{2} ( u )}{c ^{2} o ^{2}} d u

= 2 s^{3} \cdot \int \frac{u sin ^{2} ( u )}{c ^{2} o ^{2}} d u

Take the constant out:

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

= 2 s^{3} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{2} (u) d u

Apply Integration By Parts

: \frac{1}{2} u (u - \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u - \frac{1}{2} sin (2 u)) d u

= 2 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} u (u - \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u - \frac{1}{2} sin (2 u)) d u)

\int \frac{1}{2} (u - \frac{1}{2} sin (2 u)) d u = \frac{1}{2} (\frac{u ^{2}}{2} + \frac{1}{4} cos (2 u))

= 2 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} u (u - \frac{1}{2} sin (2 u)) - \frac{1}{2} (\frac{u ^{2}}{2} + \frac{1}{4} cos (2 u)))

Substitute back

u = co x

= 2 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} co x (co x - \frac{1}{2} sin (2 co x)) - \frac{1}{2} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x)))

Simplify

2 s^{3} \frac{1}{c ^{2} o ^{2}} (\frac{1}{2} co x (co x - \frac{1}{2} sin (2 co x)) - \frac{1}{2} (\frac{( co x ) ^{2}}{2} + \frac{1}{4} cos (2 co x))) : \frac{s ^{3}}{4 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x))

= \frac{s ^{3}}{4 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x))

Add a constant to the solution

= \frac{s ^{3}}{4 c ^{2} o ^{2}} (2 c^{2} o^{2} x^{2} - 2 co x sin (2 co x) - cos (2 co x)) + C

y^{^{'}} + 5 y = 2 x y^{5}

\frac{d y}{d x} + 2 x^{- 1} y = 6 x^{4} y^{2}

(\frac{1}{8}) y^{^{''}} + 4 y^{^{'}} + 20 y = 0

\int 30 (x + 9) (x + 9)^{4} d x

\frac{d}{d x} (\frac{x ^{3}}{2} - \frac{3}{x ^{2}})

What is the integral of 2sin^2(xco)s^3x ?
The integral of 2sin^2(xco)s^3x is (s^3)/(4c^2o^2)(2c^2o^2x^2-2coxsin(2cox)-cos(2cox))+C