find f. f(x) = 4e^x - 5 sin(x), f(0) = 3, f(π/2) = 0 f(x) =

Explanation:

Step1: Integrar $f''(x)$ para obtener $f'(x)$

$$f'(x)=\int(4e^{x}-5\sin(x))dx = 4e^{x}+5\cos(x)+C_1$$

Step2: Integrar $f'(x)$ para obtener $f(x)$

$$f(x)=\int(4e^{x}+5\cos(x)+C_1)dx=4e^{x}+5\sin(x)+C_1x + C_2$$

Step3: Usar la condición $f(0) = 3$

Sustituir $x = 0$ en $f(x)$: $$f(0)=4e^{0}+5\sin(0)+C_1\cdot0 + C_2=4 + C_2$$ Como $f(0)=3$, entonces $4 + C_2=3$, lo que da $C_2=- 1$.

Step4: Usar la condición $f(\frac{\pi}{2}) = 0$

Sustituir $x=\frac{\pi}{2}$, $C_2 = - 1$ en $f(x)$: $$f(\frac{\pi}{2})=4e^{\frac{\pi}{2}}+5\sin(\frac{\pi}{2})+C_1\cdot\frac{\pi}{2}-1=4e^{\frac{\pi}{2}}+5 + \frac{\pi}{2}C_1-1$$ $$0=4e^{\frac{\pi}{2}}+4+\frac{\pi}{2}C_1$$ $$\frac{\pi}{2}C_1=-4e^{\frac{\pi}{2}} - 4$$ $$C_1=-\frac{8e^{\frac{\pi}{2}} + 8}{\pi}$$

Step5: Escribir la función $f(x)$

$$f(x)=4e^{x}+5\sin(x)-\frac{8e^{\frac{\pi}{2}} + 8}{\pi}x-1$$

Answer:

$$4e^{x}+5\sin(x)-\frac{8e^{\frac{\pi}{2}} + 8}{\pi}x-1$$