Another Integration Formula

[gleem]

According to the author of this article, this integration formula is well-known but rarely taught. Do you know it?

$$\int f(x)dx = xf(x)-\int_{x_{0}}^{f(x)}f^{-1}(t)dt $$

where x₀ is a constant and f^-1(x) is the inverse function of f(x).

 

[fresh_42]

Is there a typo somewhere? I get for $f(x)=e^x$
\begin{align*}
\int e^x\,dx &\stackrel{?}{=} xe^x-\int_{x_{0}}^{e^x}\log(t)\,dt\\
&= xe^x-\left[t\log(t)-1\right]_{x_{0}}^{e^x}\\
&=xe^x-(e^x x-1)+(x_0\log(x_0)-1)\\
&=x_0\log(x_0)
&\neq e^{x}
\end{align*}
Where am I wrong?

 

[PeroK]

According to the author of this article, this integration formula is well-known but rarely taught. Do you know it?

$$\int f(x)dx = xf(x)-\int_{x_{0}}^{f(x)}f^{-1}(t)dt $$

where x₀ is a constant and f^-1(x) is the inverse function of f(x).

It's a combination of parts and substitution. You can start from the integral of an inverse function. With $f(x) = t$, do a change of variables, followed by parts.
$$\int_{t_0}^{t_1}f^{-1}(t)dt = \int_{x_0}^{x_1}xf'(x)dx$$$$ =\bigg[xf(x)\bigg]_{x_0}^{x_1} - \int_{x_0}^{x_1}f(x)dx$$

 

[PeroK]

Is there a typo somewhere? I get for $f(x)=e^x$
\begin{align*}
\int e^x\,dx &\stackrel{?}{=} xe^x-\int_{x_{0}}^{e^x}\log(t)\,dt\\
&= xe^x-\left[t\log(t)-1\right]_{x_{0}}^{e^x}\\
&=xe^x-(e^x x-1)+(x_0\log(x_0)-1)\\
&=x_0\log(x_0)
&\neq e^{x}
\end{align*}
Where am I wrong?

The integral of $\log t$ is $t\log t - t$.

 

[PAllen]

It's a combination of parts and substitution. You can start from the integral of an inverse function. With $f(x) = t$, do a change of variables, followed by parts.
$$\int_{t_0}^{t_1}f^{-1}(t)dt = \int_{x_0}^{x_1}xf'(x)dx$$$$ =\bigg[xf(x)\bigg]_{x_0}^{x_1} - \int_{x_0}^{x_1}f(x)dx$$

You can also just take the derivative of the RHS, and verify that it is f(x). I note that I have never seen this before. It looks cute, but I wouldn't know how useful it is without playing with some examples myself.

[The derivative comes out $f(x)+xf'(x)-xf'(x)$ ]

 

[Mondayman]

It looks cute, but I wouldn't know how useful it is without playing with some examples myself.

This looks like one of those integration identities that might find use on some ubsurd integral challenges (which I enjoy), but probably not much use anywhere else. I would be interested to see if it does show up in someones research somewhere.

 

[PeroK]

This looks like one of those integration identities that might find use on some ubsurd integral challenges (which I enjoy), but probably not much use anywhere else. I would be interested to see if it does show up in someones research somewhere.

It could be useful, but it doesn't have a lot of independent value. It's just the full substitution, followed by parts. In that sense, it's nothing new.