integration of x root(x4-1)

Let I = ∫xx4-1 dx=∫xx22-1 dxput x2 = t⇒2x dx = dt⇒x dx = dt2So, I = 12∫t2-1=12t2t2-1 - 12log t+t2-1 + C=12x22x4-1 - 12logx2+x4-1 + C

integration of x root(x⁴-1)

Let I = \int x \sqrt{x^{4} - 1} dx = \int x \sqrt{{(x^{2})}^{2} - 1} dx put x^{2} = t \Rightarrow 2 x dx = dt \Rightarrow x dx = \frac{dt}{2} So, I = \frac{1}{2} \int \sqrt{t^{2} - 1} = \frac{1}{2} [\frac{t}{2} \sqrt{t^{2} - 1} - \frac{1}{2} \log |t + \sqrt{t^{2} - 1}|] + C = \frac{1}{2} [\frac{x^{2}}{2} \sqrt{x^{4} - 1} - \frac{1}{2} \log |x^{2} + \sqrt{x^{4} - 1}|] + C