What is the integral of 4^xsin(4^x) ?

The integral of 4^xsin(4^x) is -(4^xcos(4^x))/(ln(2)*2^{2x+1)}+C

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integral of 4^xsin(4^x)

\int 4^{x} sin (4^{x}) d x

- \frac{4 ^{x} cos ( 4 ^{x} )}{ln ( 2 ) \cdot 2 ^{2 x + 1}} + C

Solution steps

\int 4^{x} sin (4^{x}) d x

Apply u-substitution

: \int \frac{u sin ( u )}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} d u

= \int \frac{u sin ( u )}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} d u

Take the constant out:

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

= \frac{1}{ln ( 2 )} \cdot \int \frac{u sin ( u )}{2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} d u

Apply Integration By Parts

: \frac{- u cos ( u ) + sin ( u )}{2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} - \int - \frac{- u cos ( u ) + sin ( u )}{2 u ^{2}} d u

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

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

= \frac{1}{ln ( 2 )} (\frac{- u cos ( u ) + sin ( u )}{2 ^{\frac{l n ( u ) + l n ( 2 )}{l n ( 2 )}}} - \frac{sin ( u )}{2 u})

Substitute back

u = 4^{x}

= \frac{1}{ln ( 2 )} (\frac{- 4 ^{x} cos ( 4 ^{x} ) + sin ( 4 ^{x} )}{2 ^{\frac{l n ( 4 ^{x} ) + l n ( 2 )}{l n ( 2 )}}} - \frac{sin ( 4 ^{x} )}{2 \cdot 4 ^{x}})

Simplify

\frac{1}{ln ( 2 )} (\frac{- 4 ^{x} cos ( 4 ^{x} ) + sin ( 4 ^{x} )}{2 ^{\frac{l n ( 4 ^{x} ) + l n ( 2 )}{l n ( 2 )}}} - \frac{sin ( 4 ^{x} )}{2 \cdot 4 ^{x}}) : - \frac{4 ^{x} cos ( 4 ^{x} )}{ln ( 2 ) \cdot 2 ^{2 x + 1}}

= - \frac{4 ^{x} cos ( 4 ^{x} )}{ln ( 2 ) \cdot 2 ^{2 x + 1}}

Add a constant to the solution

= - \frac{4 ^{x} cos ( 4 ^{x} )}{ln ( 2 ) \cdot 2 ^{2 x + 1}} + C

x \to 0 + lim (sin (x) ln (4 x))

\int t cos (n t) d t

k \to \infty lim (\frac{k ^{\frac{1}{k}}}{ln ( k )})

\frac{d}{d s} (\frac{s ^{2} - 4 s + 20}{( s + 2 ) ( s + 4 )})

\int e^{t a n (6 x)} sec^{2} (6 x) d x

What is the integral of 4^xsin(4^x) ?
The integral of 4^xsin(4^x) is -(4^xcos(4^x))/(ln(2)*2^{2x+1)}+C