What is the integral of (sin(2x)+(x-2)/x-4^x) ?

The integral of (sin(2x)+(x-2)/x-4^x) is -1/2 cos(2x)+x-2ln|x|-(2^{2x-1})/(ln(2))+C

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integral of (sin(2x)+(x-2)/x-4^x)

\int (sin (2 x) + \frac{x - 2}{x} - 4^{x}) d x

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

Solution steps

\int sin (2 x) + \frac{x - 2}{x} - 4^{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 sin (2 x) d x + \int \frac{x - 2}{x} d x - \int 4^{x} d x

\int sin (2 x) d x = - \frac{1}{2} cos (2 x)

\int \frac{x - 2}{x} d x = x - 2 ln ∣ x ∣

\int 4^{x} d x = \frac{2 ^{2 x - 1}}{ln ( 2 )}

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

Add a constant to the solution

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

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What is the integral of (sin(2x)+(x-2)/x-4^x) ?
The integral of (sin(2x)+(x-2)/x-4^x) is -1/2 cos(2x)+x-2ln|x|-(2^{2x-1})/(ln(2))+C