Question

integral :3^x * cosx.dx

Answer 1

$\displaystyle I=\int 3^x \cos x \;dx\\ \\ I=\int e^{x \ln 3} \cos x \;dx\\ \\ I=\dfrac{1}{\ln 3}\int \cos x \;d(e^{x \ln 3})\\ \\ I=\dfrac{1}{\ln 3}\left(e^{x \ln 3}\cos x - \int e^{x \ln 3} \;d(\cos x)\right)\\ \\ I=\dfrac{1}{\ln 3}\left(e^{x \ln 3}\cos x +\int e^{x \ln 3}\sin x \;dx\right)$

$\displaystyle I=\dfrac{1}{\ln 3}\left(e^{x \ln 3}\cos x +\dfrac{1}{\ln 3}\int \sin x \;d(e^{x \ln 3})\right)\\ \\ I=\dfrac{1}{\ln 3}\left[e^{x \ln 3}\cos x +\dfrac{1}{\ln 3}\left(e^{x \ln 3}\sin x -\int e^{x \ln 3}\;d(\sin x)\right)\right]\\ \\ I=\dfrac{1}{\ln 3}\left[e^{x \ln 3}\cos x +\dfrac{1}{\ln 3}\left(e^{x \ln 3}\sin x -\int e^{x \ln 3}\cos x\;dx\right)\right]$

$I=\dfrac{1}{\ln 3}\left[e^{x \ln 3}\cos x +\dfrac{1}{\ln 3}\left(e^{x \ln 3}\sin x -I\right)\right]\\ \\ \\ I=\dfrac{1}{\ln 3}e^{x \ln 3}\cos x +\dfrac{1}{\ln^2 3}\,e^{x \ln 3}\sin x -Ie^{x \ln 3}\\ \\ \\ (1+e^{x \ln 3})I=\dfrac{1}{\ln 3}3^x \cos x +\dfrac{1}{\ln^2 3}\,3^x \sin x\\ \\ \\ \boxed{I=\dfrac{3^x(\cos x \ln 3+\sin x)}{(1+3^x)\ln^2 3} +C}$