Question

integral x.e^x. dx please help​

Answer 1

Solution:-

Using Integration by part method

$\rm \int(UV)dx = U \int Vdx - \int \bigg \{ \bigg( \dfrac{du}{dx} \bigg) \int vdx \bigg \}dx$

We have to integrate

$\to \rm\int x {e}^{x} dx$

Now determine the U and V use this method

=> I L A T E

:- I = inverse trigonometry

:- L = Logarithmic

:- A = Algebra

:- T = Trigonometry

:- E = Exponential

Now according to this method take

=> x = U and e^x = V

Now we can write

$\to \rm\int x {e}^{x} dx$

$\rm \: = x \int e {}^{x} dx - \int \bigg \{ \bigg( \dfrac{d}{dx} x \bigg) \int e{}^{x} dx \bigg \}dx + c$

$\rm \: = x \times e {}^{x} - \int(1) \{\times e {}^{x} \big \}dx + c$

$\rm = x {e}^{x} - \int {e}^{x} dx + c$

$= \rm \: x {e}^{x} - {e}^{x} + c$

Answer

$= \rm \: x {e}^{x} - {e}^{x} + c$