Q115 / 160 · TS EAMCET 2024 (11 May Shift 1) · Mathematics

If $\int e^x\left(x^3+x^2-x+4\right) d x=e^x f(x)+c$, then $f(1)=$

  • A
    $0$
  • B
    $1$
  • C
    $2$
  • Correct answer
    D
    $3$

Solution

$\begin{aligned} & \int e^x\left(x^3+x^2-x+4\right) d x=e^x f(x)+\mathrm{c} \\ \Rightarrow & e^x\left(x^3+x^2-x+4\right)=e^x f^{\prime}(x)+e^x f(x) \\ \Rightarrow & f(x)+f^{\prime}(x)=x^3+x^2-x+4 \end{aligned}$ $\begin{aligned} & \text { Let } f(x)=x^3+a x^2+b x+c \Rightarrow f^{\prime}(x)=3 x^2+2 a x+b \\ & \Rightarrow x^3+x^2-x+4=x^3+(a+3) x^2+(2 a+b) x+b+c \\ & \Rightarrow a+3=1 \Rightarrow a=-2 \Rightarrow 2 a+b=-1 \Rightarrow b=3 \\ & b+c=4 \Rightarrow c=1 \therefore f(x)=x^3-2 x^2+3 x+1 \\ & f(1)=1-2+3+1=3 \end{aligned}$

Hence, the correct answer is option: 'D'.

Relevant Questions