Answer to Question #286831 in Calculus for Moon

Question

$\intop$₂⁴ e^x cosxdx=?

Answer 1

$$\int e^x\cos xdx=e^x\sin x-\int e^x\sin xdx$$

$$=e^x\sin x-(-e^x\cos x+\int e^x\cos xdx)$$

$$=e^x\sin x+e^x\cos x-\int e^x\cos x dx$$

$$\int e^x\cos xdx=\dfrac{1}{2}(e^x\sin x+e^x\cos x)+C$$

$$\displaystyle\int_{2}^{4}e^x\cos xdx=[\dfrac{1}{2}(e^x\sin x+e^x\cos x)]\begin{matrix} 4 \\ 2 \end{matrix}$$

$$=\dfrac{1}{2}(e^4\sin 4+e^4\cos 4-e^2\sin 2-e^2\cos 2)$$