Integral that evaluates to 42

Question

I know this isn't a specific math question, but I need some help coming up with an integral that evaluates to 42 for a calc shirt. Obviously, I'm striving for something more complicated than $\int_{10}^{11} 4x dx$ or $\int_{0}^{42} 1 dx$. Try and stick to integral content that would be covered in the BC calc curriculum, because that is the class I'm trying to design the shirt for. Thanks!

Answer 1

A nice and simple one is $$\int_0^\infty (2x^4-x^3)e^{-x}\, dx$$ This works, since for any natural number $n$, $$\int_0^{\infty} x^ne^{-x}\,dx = n!$$

Answer 2

You can try: \begin{align} \int^4_0 \frac{945 } {2(x^2+9)^{3/2} } dx \end{align}

Solvable by suitable substitution $x=a\tan z$. Or you can try: \begin{align} \int_0^{2\pi}\frac{63 }{5\pi+4\pi\cos x}dx \end{align} Solvable by $z=\tan(x/2)$

I think both techniques is taught in BC Calculus.

Answer 3

Partial fractions for this one... $\int_1^2 \frac{336}{\ln(\frac{5}{2}) x(x^2+4)} dx \\\ $

I cheated...I played with the constant multiples to get 42 for the answer to the definite integral. :p

Answer 4

Because $42$ is the fifth Catalan number then using the known integral representation of the Catalan's numbers we have that

$$C_5=\frac1{2\pi}\int_0^4x^5\sqrt{\frac{4-x}x}\,\mathrm dx=42$$

Playing with some change of variable above you can get different integral expressions. By example with the change of variable $(4-x)/x=t$ we can build the following improper integral

$$C_5=\frac{16}{\pi}\int_0^\infty\sqrt t\left(\frac12+\frac{t}2\right)^{-7}\mathrm dt=42$$

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Maybe more interesting integral representations can be achieved considering first the integral representations for $21$, what is a number with more interesting properties.