How To Integrate sin^-1(x)
Published On :2020-12-06 20:47:00
[begin{array}{l} smallint {sin ^{ - 1}}x{rm{d}}x\\ ByParts\ u = si{n^{( - 1)}}x;;;;;;;;;;;;;;;{rm{d}}u = frac{{dx}}{{sqrt {1 - {x^2}} }}\ dv = 1;;;;;;;;;;;;;;;;;;;;;;;;v = x\\ x{sin ^{ - 1}}x - smallint x cdot frac{1}{{sqrt {1 - {x^2}} }}{rm{d}}x\ By{rm{ }}Substitution;;;;;;;;;;;;;\ ;;;Suppose Rightarrow z = {x^2};;;;;;;;;;;;;;;;;\ ;;;;;;;;;;;;;;;;;;;;dz = 2;x;dx\\ x{sin ^{ - 1}}x - frac{1}{2}smallint frac{1}{{sqrt {1 - z} }}{rm{d}}z\ Rightarrow x{sin ^{ - 1}}x + frac{1}{2} cdot 2{left( {1 - z} right)^{frac{1}{2}}} + c;\ Rightarrow x{sin ^{ - 1}}x + sqrt {1 - {x^2}} + c end{array}]