integration of xlnx/(x2-1)3/2 dx

Dear student, ∫xlnxx2-132dx=lnx2∫2xx2-132dx-∫ddxlnx2∫2xx2-132dxdxlet x2-1=t, 2xdx=dt=lnx2∫1t32dt-∫ddxlnx2∫1t32dtdx=lnx2-2t12-∫12x-2t12dx=-lnxx2-112+∫1x1x2-112dx=-lnxx2-112+sec-1x+c Regards

integration of xlnx/(x²-1)^3/2 dx

Dear student,

\int \frac{x \ln x}{{(x^{2} - 1)}^{\frac{3}{2}}} d x = \frac{\ln x}{2} \int \frac{2 x}{{(x^{2} - 1)}^{\frac{3}{2}}} d x - \int [(\frac{d}{d x} (\frac{\ln x}{2})) \int \frac{2 x}{{(x^{2} - 1)}^{\frac{3}{2}}} d x] d x l e t x^{2} - 1 = t, 2 x d x = d t = \frac{\ln x}{2} \int \frac{1}{{(t)}^{\frac{3}{2}}} d t - \int [(\frac{d}{d x} (\frac{\ln x}{2})) \int \frac{1}{{(t)}^{\frac{3}{2}}} d t] d x = \frac{\ln x}{2} \frac{- 2}{{(t)}^{\frac{1}{2}}} - \int [(\frac{1}{2 x}) \frac{- 2}{{(t)}^{\frac{1}{2}}}] d x = \frac{- \ln x}{{(x^{2} - 1)}^{\frac{1}{2}}} + \int \frac{1}{x} \frac{1}{{(x^{2} - 1)}^{\frac{1}{2}}} d x = \frac{- \ln x}{{(x^{2} - 1)}^{\frac{1}{2}}} + s e c^{- 1} x + c

Regards

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integration of xlnx​​​/(x2-1)3/2 dx

integration of xlnx/(x²-1)^3/2 dx