Integration Strategy — Question 6

Question 6 – Solution

Step 1: Test the proposed substitution. If $$u=x^2+1,$$ then $$du=2x\,dx.$$ The integral contains $dx$, but it does not contain the required factor $x\,dx$. Thus the substitution does not produce a direct $du$ factor. One can express $dx$ in terms of $u$ on a branch, but this introduces a square root and is less convenient than the standard antiderivative.

Step 2: Recognize a standard form. $$\frac{d}{dx}(\arctan x)=\frac1{1+x^2}.$$ Therefore, $$\int\frac{dx}{x^2+1}=\arctan x+C.$$ $$\boxed{\int\frac{dx}{x^2+1}=\arctan x+C}$$