What is the integral of 2^{sin^2(1/3)x}sin(2/3)x ?

The integral of 2^{sin^2(1/3)x}sin(2/3)x is

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integral of 2^{sin^2(1/3)x}sin(2/3)x

\int 2^{s i n^{2} (\frac{1}{3}) x} sin (\frac{2}{3}) x d x

\frac{csc ^{2} ( \frac{1}{3} ) sin ( \frac{2}{3} ) \cdot 2 ^{s i n^{2} (\frac{1}{3}) x} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} sin (\frac{2}{3}) \cdot 2^{s i n^{2} (\frac{1}{3}) x} + C

Solution steps

\int 2^{s i n^{2} (\frac{1}{3}) x} sin (\frac{2}{3}) x d x

Take the constant out:

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

= sin (\frac{2}{3}) \cdot \int 2^{s i n^{2} (\frac{1}{3}) x} x d x

Apply u-substitution

: \int \frac{csc ^{2} ( \frac{1}{3} ) \cdot 2 ^{u} u}{sin ^{2} ( \frac{1}{3} )} d u

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{2 ^{u} u}{ln ( 2 )} - \int \frac{2 ^{u}}{ln ( 2 )} d u

= sin (\frac{2}{3}) \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} (\frac{2 ^{u} u}{ln ( 2 )} - \int \frac{2 ^{u}}{ln ( 2 )} d u)

\int \frac{2 ^{u}}{ln ( 2 )} d u = \frac{2 ^{u}}{ln ^{2} ( 2 )}

= sin (\frac{2}{3}) \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} (\frac{2 ^{u} u}{ln ( 2 )} - \frac{2 ^{u}}{ln ^{2} ( 2 )})

Substitute back

u = sin^{2} (\frac{1}{3}) x

= sin (\frac{2}{3}) \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} (\frac{2 ^{s i n^{2} (\frac{1}{3}) x} sin ^{2} ( \frac{1}{3} ) x}{ln ( 2 )} - \frac{2 ^{s i n^{2} (\frac{1}{3}) x}}{ln ^{2} ( 2 )})

Simplify

sin (\frac{2}{3}) \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} (\frac{2 ^{s i n^{2} (\frac{1}{3}) x} sin ^{2} ( \frac{1}{3} ) x}{ln ( 2 )} - \frac{2 ^{s i n^{2} (\frac{1}{3}) x}}{ln ^{2} ( 2 )}) : \frac{csc ^{2} ( \frac{1}{3} ) sin ( \frac{2}{3} ) \cdot 2 ^{s i n^{2} (\frac{1}{3}) x} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} sin (\frac{2}{3}) \cdot 2^{s i n^{2} (\frac{1}{3}) x}

= \frac{csc ^{2} ( \frac{1}{3} ) sin ( \frac{2}{3} ) \cdot 2 ^{s i n^{2} (\frac{1}{3}) x} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} sin (\frac{2}{3}) \cdot 2^{s i n^{2} (\frac{1}{3}) x}

Add a constant to the solution

= \frac{csc ^{2} ( \frac{1}{3} ) sin ( \frac{2}{3} ) \cdot 2 ^{s i n^{2} (\frac{1}{3}) x} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot \frac{csc ^{2} ( \frac{1}{3} )}{sin ^{2} ( \frac{1}{3} )} sin (\frac{2}{3}) \cdot 2^{s i n^{2} (\frac{1}{3}) x} + C

\frac{d}{d x} (x y e^{- x^{2} - y^{2}})

\frac{d}{d x} (e^{(- 4 x)})

y^{^{'}} + y = \frac{1}{1 + e ^{x}}

\frac{d}{d x} (sin (x^{0}))

What is the integral of 2^{sin^2(1/3)x}sin(2/3)x ?
The integral of 2^{sin^2(1/3)x}sin(2/3)x is