Question

integral e^x(x^3+4x^2 +3x+4 )dx​

Answer 1

We recall,

$\displaystyle\longrightarrow\int e^x\left[f(x)+f'(x)\right]\ dx=e^x f(x)+C\quad\quad\dots(1)$

We are given,

$\displaystyle\longrightarrow I=\int e^x\left[x^3+4x^2+3x+4\right]\,dx$

Split the polynomial in such a way that each term is followed by it's first derivative wrt $x.$

$\displaystyle\longrightarrow I=\int e^x\left[x^3+3x^2+x^2+2x+x+1+3\right]\,dx$

Splitting the integral as follows.

$\displaystyle\longrightarrow I=\int e^x\left[x^3+3x^2\right]\,dx+\int e^x\left[x^2+2x\right]\,dx+\int e^x\left[x+1\right]\,dx+3\int e^x\,dx$

Using (1),

$\displaystyle\longrightarrow I=x^3\,e^x+x^2\,e^x+x\,e^x+3e^x+C$

$\displaystyle\longrightarrow\underline{\underline{I=e^x(x^3+x^2+x+3)+C}}$