Question

lf $f\left(x\right)$ is a polynomial of nth degree then $\int e^{x}f\left(x\right)dx=$

• A. $e^x\left[f\right(x)-f^{{}^{\prime }}(x)+f^{{}^{\prime \prime }}(x)-f^{{}^{\prime \prime \prime }}(x)+\dots \dots +(-1{)}^n{f}^n\left(x\right)]$ Where $f^n\left(x\right)$ denotes nth order derivative of w.r.t. $x$
• B. $e^x\left[f\right(x)+f(x)+f^{{}^{\prime }}(x)+f^{{}^{\prime \prime }}(x)+\dots \dots +(-1{)}^n{J}^n\left(x\right)]$
• C. $e^x\left[f\right(x)+f^{{}^{\prime }}(x)+f^{{}^{\prime }}(x)+f^{{}^{\prime \prime \prime }}(x)+\dots \dots +(-1{)}^n{f}^{2n}\left(x\right)]$
• D. $d\left[f\right(x)+f(x)+f^{{}^{\prime }}(x)+f^{{}^{\prime \prime \prime }}(x)+\dots \dots +\{-1{)}^n{f}^{3n}\left(x\right)]$

Answer 1

The correct option is A $e^x\left[f\right(x)-f^{{}^{\prime }}(x)+f^{{}^{\prime \prime }}(x)-f^{{}^{\prime \prime \prime }}(x)+\dots \dots +(-1{)}^n{f}^n\left(x\right)]$ Where $f^n\left(x\right)$ denotes nth order derivative of w.r.t. $x$

$\int e^{x}f\left(x\right)dx$
$=f\left(x\right).e^x-\int f^{\prime }\left(x\right)e^x$
$=f\left(x\right).e^x-f^{\prime }\left(x\right)e^x+\int f^{\prime \prime }\left(x\right)e^x$
$=f\left(x\right).e^x-f^{\prime }\left(x\right)e^x+f^{\prime \prime }\left(x\right)e^x-\int f^{\prime \prime \prime }\left(x\right)e^x$
$=e^x\left[f\right(x)-f^{\prime }(x)+f^{\prime \prime }(x).......(-1{)}^n{f}^n\left(x\right)]$