Use the substitution x=tan(θ) to evaluate ∫ (x dx)/(√ x^2 +1)

Answer 1

Let x = tanθ. Then, dx = sec²θdθ

So, ∫[x / √(x² + 1)]dx = ∫[tanθ/√(tan²θ+1) sec²θ]dθ = ∫[tanθsecθdθ = secθ + C

Draw a right triangle with acute angle θ, side opposite θ of length x and the other leg of length 1 so that

tanθ = x. Then, by the Pythagorean Theorem, the hypotenuse is √[1+x²].

So, secθ + C = √[1+x²] + C