What is the integral of cos(x)ln(2)2^x+2^xsin(x) ?

The integral of cos(x)ln(2)2^x+2^xsin(x) is

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integral of cos(x)ln(2)2^x+2^xsin(x)

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

(\frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{ln ( 2 ) + ln ^{3} ( 2 )} + \frac{ln ^{2} ( 2 ) \cdot 2 ^{x} cos ( x )}{ln ( 2 ) + ln ^{3} ( 2 )}) ln (2) - \frac{2 ^{x} cos ( x )}{1 + ln ^{2} ( 2 )} + \frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{1 + ln ^{2} ( 2 )} + C

Solution steps

\int cos (x) ln (2) \cdot 2^{x} + 2^{x} sin (x) d x

Apply the Sum Rule:

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

= \int cos (x) ln (2) \cdot 2^{x} d x + \int 2^{x} sin (x) d x

\int cos (x) ln (2) \cdot 2^{x} d x = (\frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{ln ( 2 ) + ln ^{3} ( 2 )} + \frac{ln ^{2} ( 2 ) \cdot 2 ^{x} cos ( x )}{ln ( 2 ) + ln ^{3} ( 2 )}) ln (2)

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

= (\frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{ln ( 2 ) + ln ^{3} ( 2 )} + \frac{ln ^{2} ( 2 ) \cdot 2 ^{x} cos ( x )}{ln ( 2 ) + ln ^{3} ( 2 )}) ln (2) - \frac{2 ^{x} cos ( x )}{1 + ln ^{2} ( 2 )} + \frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{1 + ln ^{2} ( 2 )}

Add a constant to the solution

= (\frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{ln ( 2 ) + ln ^{3} ( 2 )} + \frac{ln ^{2} ( 2 ) \cdot 2 ^{x} cos ( x )}{ln ( 2 ) + ln ^{3} ( 2 )}) ln (2) - \frac{2 ^{x} cos ( x )}{1 + ln ^{2} ( 2 )} + \frac{ln ( 2 ) \cdot 2 ^{x} sin ( x )}{1 + ln ^{2} ( 2 )} + C

\frac{d}{d x} ((3 x - 2)^{2} (5 x + 4))

\int (8 x + 3) (7 - x) d x

x \to - \infty lim (- 3 x + x)

h = (t + 1)^{\frac{2}{3}} (2 t^{2} - 1)^{3}

\int \frac{x x}{2} d x

What is the integral of cos(x)ln(2)2^x+2^xsin(x) ?
The integral of cos(x)ln(2)2^x+2^xsin(x) is