What is the integral of 3^xcos(x) ?

The integral of 3^xcos(x) is (3^xsin(x))/(1+ln^2(3))+(ln(3)*3^xcos(x))/(1+ln^2(3))+C

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integral of 3^xcos(x)

\int 3^{x} cos (x) d x

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

Solution steps

\int 3^{x} cos (x) d x

Apply Integration By Parts

: 3^{x} sin (x) - \int 3^{x} ln (3) sin (x) d x

= 3^{x} sin (x) - \int 3^{x} ln (3) sin (x) d x

Take the constant out:

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

= 3^{x} sin (x) - ln (3) \cdot \int 3^{x} sin (x) d x

Apply Integration By Parts

: - 3^{x} cos (x) - \int - ln (3) \cdot 3^{x} cos (x) d x

= 3^{x} sin (x) - ln (3) (- 3^{x} cos (x) - \int - ln (3) \cdot 3^{x} cos (x) d x)

Take the constant out:

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

= 3^{x} sin (x) - ln (3) (- 3^{x} cos (x) - (- ln (3) \cdot \int 3^{x} cos (x) d x))

Therefore

\int 3^{x} cos (x) d x = 3^{x} sin (x) - ln (3) (- 3^{x} cos (x) - (- ln (3) \cdot \int 3^{x} cos (x) d x))

Isolate

\int 3^{x} cos (x) d x

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

Add a constant to the solution

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

x \to \infty lim (x e^{\frac{1}{x}})

\int cos (y) sin (x) d x

\int x 3 x^{2} + 1 d x

x \to 0 lim (\frac{sin ( 9 x )}{x})

(2 x y + 3 y^{2}) d x - (2 x y + x^{2}) d y = 0

What is the integral of 3^xcos(x) ?
The integral of 3^xcos(x) is (3^xsin(x))/(1+ln^2(3))+(ln(3)*3^xcos(x))/(1+ln^2(3))+C