What is the integral of 2^x*cos(x) ?

The integral of 2^x*cos(x) is (2^xsin(x))/(1+ln^2(2))+(ln(2)*2^xcos(x))/(1+ln^2(2))+C

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integral of 2^x*cos(x)

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

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

Solution steps

\int 2^{x} cos (x) d x

Apply Integration By Parts

: 2^{x} sin (x) - \int 2^{x} ln (2) sin (x) d x

= 2^{x} sin (x) - \int 2^{x} ln (2) sin (x) d x

Take the constant out:

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

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

Apply Integration By Parts

: - 2^{x} cos (x) - \int - ln (2) \cdot 2^{x} cos (x) d x

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

Take the constant out:

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

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

Therefore

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

Isolate

\int 2^{x} cos (x) d x

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

Add a constant to the solution

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

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