# Difficult integrals?

I'm looking for some tricky/difficult integrals within the scope of calc I and II that I can play around with. Most of the integrals in my books (Stewart and Spivak) are fairly straight forward, and the only real practice I get is in "rigor". I can't really make up my own problems either, because I always come up with something unsolvable (without a CAS et al).

What are some good integrals??

What are some good integrals??

You may hit the SEARCH of this forum with 'integrals'.

https://www.physicsforums.com/showpost.php?p=3433157&postcount=272
$$\int \sin(\ln x) + \cos(\ln x)dx$$
$$\int \frac{x^2}{x^2 +4x + 8} dx$$
$$\int \frac{1}{\sqrt{5x-3}+\sqrt{5x+2}} dx$$
$$\int \left( x^2 + 1\right) e^{x^2}dx$$
$$\int \frac{1}{\sqrt[3]{x} + x} dx$$
The integral below is tricky, BUT it can be solved using only simple substitutions.
Show that

$$I_4 \, = \, \int_{0}^{\infty} \dfrac{x^{29}}{(5x^2+49)^{17}} \, dx \,=\, \dfrac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}$$

What I like about these integrals, is that most of them have simple, clever solutions.

Last edited:
micromass
Staff Emeritus
$$\int\frac{4x^5-1}{(x^5+x+1)^2}dx$$