[Solved] If $\int E^x (f(x) - F'(x))dx = \phi(x),$ Then T

Có thể bạn quan tâm

Calculation:

Here, we have to find the value of the integrand $\int e^x (f(x) - f'(x))dx = \phi(x)$

$\int e^xf'(x))dx =- \phi(x)+\int e^x f(x)dx$ --------------(1)

For given Integration $\int e^x f(x)dx$ according to ILATE rule u = f(x); v = ex

$\int {{e^x}} f(x)dx = f(x)\int {{e^x}dx} - \int {\left[ {\frac{{df(x)}}{{dx}}\int {{e^x}dx} } \right]} dx$

$\int {{e^x}} f(x)dx = f(x){e^x} - \int {f'(x){e^x}} dx$

$\int {{e^x}} f(x)dx = {e^x}f(x) - \int {{e^x}f'(x)} dx$

Put the value of $\int {{e^x}f'(x)} dx$ from equation (1)

$\int {{e^x}} f(x)dx = {e^x}f(x) + \phi (x) - \int {{e^x}f(x)} dx$

$2\int {{e^x}} f(x)dx = {e^x}f(x) + \phi (x)$

$\int {{e^x}} f(x)dx = \frac{1}{2}[{e^x}f(x) + \phi (x)]$

Download Solution PDF Share on Whatsapp

Từ khóa » Dx Phi

[Solved] If \(\int E^x (f(x) - F'(x))dx = \phi(x),\) Then T