Is it normal for integration to be a slow process?

Question

I have currently been doing integration problems, and it seems to me that I am quite slow at the process.

I was wondering whether that was normal or am I using inefficient methods to get to my final solution?

For example integrating: $\frac{x}{e^{x}}$ takes me about 10 to 12 minutes. I use a combination of integration by parts and integration by substitution to get my answer.

This is my working out process:

$u = x$

$du = 1 dx $

$v = -e^{-x}$ (Used integration by substitution)

$dv = e^{-x} dx$

Integration by Parts form: $uv - \int u \, du$

$-xe^{-x} - \int-e^{-x} dx$

** For the integration section I used Integration by parts.

Final Answer:

$(-e^{-x})(x+1)+C$

Answer 1

You would express $du$ and $dv$ if you were to compute e.g. $\int \sin^2(x)\cos(x)dx$. You could say $u=\sin(x)$, that is implicitly $u(x)=\sin(x)$. Then you have $u'(x)=\sin'(x)=\cos(x)$, or $du/dx=\sin'(x)=\cos(x)$. That way, you can rearrange to $dx=du/\cos(x)$. And $\int \sin^2(x)\cos(x)dx = \int u^2\cos(x)\frac{du}{\cos(x)}= \int u^2du$ and there you go. If you're just looking for ANY antiderivative, no worries about the boundaries changing on the integral. Otherwise that needs your attention too.

This is an example, this integral is actually easier when noticing that you have $(f\circ g)'(x)$ inside it with $f(x)=\frac{x^3}{3}$ and $g(x)=\sin(x)$

Answer 2

I think there's a problem : when assigning $u$ and $v$ in preperation for integration by parts, you are not using variable substitution. There's no need here to express $du$ and $dv$ in terms of $dx$. I'm not saying you can never use both methods but I think you got them confused here, didn't you ?

Assuming you were integrating $xe^{-x}$, a sensible choice would be $u(x)=x$ and $v'(x)=e^{-x}$, because your integral is now written as $\int u(x)v'(x)dx$, hence when using the $[u(x)v(x)]-\int u'(x)v(x)dx$ formula this integral is easy to calculate as $u'(x)=1$.

If I was right about your confusion, then it's probably due to common but incorrect use of notations. When integrating by parts, you are merely rewriting the integral without changing the integrated domain nor the integrated function. You are just spotting a clever product inside the integral to be computed, and rewriting it for visual comfort when applying the formula. Like you might write $u=x^2$ and $v=\sin(x)$ to apply the $(uv)'=u'v+uv'$ formula to $x^2\sin(x)$