Question

integral de 3^x*cos(x)dx

Answer 1

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

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