The term 1/(a2 + x2) is purely algebraic. However, its integral is trigonometric. When you look this up in a table of integrals, you'll find the integral:
∫1/(a2+x2)dx = (1/a)arctan(x/a) + C
How is this so?
This is where Pythagora's Theorem and the rules of trigonometric come in to help.
We can't simply use a u-substitution to solve this problem. Instead, if we construct a right-angle triangle with the 2 shorter sides of lengths a and x, then the relationship between the hypotenuse and the the 2 shorter sides is a2+x2 (Pythagoras' Theorem).
Now tanθ = (x/a), or x = atanθ.
Taking the derivative with respect to θ, we get dx/dθ = sec2θ.
Thus we need to substitute x = atanθ, and dx = sec2θdθ to solve the integral.
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