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Let $f(x)= e^x + 2x + 1$ and $g(x)$ be inverse of $f(x)$.
Find $\int_0^{e+3}g(x) dx$

My attempt: I tried to find the inverse of $f(x)$ but was not able to do it. Then I switched to method of area. But don't know how to proceed. I know that if $A$ is the area of $f(x)$ with x-axis then the area of $g(x)$ with y-axis would be equal to $A$. But I'm not able to find the corresponding limits for the inverse function to integrate.

cshubhamrao
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Vidit Kulshreshtha
  • 35

1 Answers1

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By the change of variable $x=e^y+2y+1$, giving $dx=(e^y+2)\,dy$, we have

$$\int_0^{e+3}g(x)\,dx=\int_{g(0)}^1y(e^y+2)\,dy=\left.((y-1)e^y+y^2)\right|_{g(0)}^1.$$