tan(θ) = x 3 Video Example EXAMPLE 3 Find 1 x2 x2 + 9 dx . SOLUTION Let x = 3 tan(θ), −π/2 < θ < π/2. Then dx = dθ and x2 + 9 = 9 tan2(θ) + 9 = 9 sec2(θ) = 3|sec(θ)| = 3 sec(θ). Thus we have dx x2 x2 + 9 = dθ 9 tan2(θ) · 3 sec(θ) = 1 9 tan2(θ) dθ. To evaluate this trigonometric integral we put everything in terms of sin(θ) and cos(θ). sec(θ) tan2(θ) = 1 · cos2(θ) sin2(θ) = sin2(θ) Therefore, making the substitution u = sin(θ), we have dx x2 x2 + 9 = 1 9