List of interesting integrals for early calculus students

Solution 1:

You might consider the old warhorse $$ \int \sec x\ dx $$ It's very common in calculus texts to resort to the trick of multiplying and dividing by $(\sec x + \tan x)$, upon doing which the answer jumps right out with a bit of simplification. Any reasonable student, though, might complain about this "rabbit out of the hat approach," asking, "How on earth could you expect me to come up with this idea?" All this approach does is impress the student with the author's cleverness while at the same time making them feel stupid. Here's an alternative approach that involves a different, and perhaps more accessible, kind of cleverness.

$$ \begin{align} \int\sec x\ dx&=\int\frac{1}{\cos x}\ dx=\int \frac{\cos x}{\cos^2 x}\ dx = \int \frac{\cos x}{1-\sin^2 x}\ dx\\ &= \int \cos x\left(\frac{1}{(1-\sin x)(1+\sin x)}\right)\ dx\\ \end{align} $$ Continue with partial fractions: $$ \begin{align} &=\int \frac{\cos x}{2}\left(\frac{1}{1-\sin x}+\frac{1}{1+\sin x}\right)\ dx\\ &= \frac{1}{2}\int\frac{\cos x}{1-\sin x}\ dx+\frac{1}{2}\int \frac{\cos x}{1+\sin x}\ dx\\ \end{align} $$ and now two simple substitutions and a bit of algebra gives the result. Occasionally, after giving this version I'll give the textbook version as an exercise, where it properly belongs.

Solution 2:

One pair of integrals they might find interesting is $$\int_0^{\pi/2} \cos^2 x \, dx \textrm{ and } \int_0^{\pi/2} \sin^2 x \, dx.$$

These integrals can be evaluated two different ways.

1. Use double angle formulas to find the antiderivatives.

2. Intuitively, the integrals should be the same, because they're the same function only flipped around. More formally, your students can check that if you make the substitution $u=\frac{\pi}{2}-x$ it turns one integral into the other. But their sum is $\int_0^{\pi/2} \sin^2 x + \cos^2 x \, dx=\int_0^{\pi/2} 1 \, dx$.

By the same trick, you can have your students integrate $$\int_0^{\pi/2} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} dx$$

Solution 3:

I remember spending a lot of time trying to crack $$\int \frac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx$$ back in the day, which became much simpler when I found out one could just let $u^6=x$. For your Calc 2 class, I've always been fond of $\int \sqrt{\tan{x}}\,dx$. It uses almost the whole cornucopia of tricks (substitution, completing the square, partial fractions).

Also, sometimes integrals which one might normally approach with trig substitution are much quicker if one knows about explicit formulae for inverse hyperbolic trig functions. For example, $$ \int\frac{1}{\sqrt{x^2+1}}\,dx$$ can be done with a trig substitution, or by noticing this is $\mathrm{arsinh}(x)+c$. In any case, you get $\log{(x+\sqrt{1+x^2}})+ c$, but it just depends whether you'd rather memorize the inverse trig formulae or do the trig subs.