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Example3: Find ò  ——————— . dx

                                       ———

                                x³ \/q² + x²  dx

Let dx = – q cosechθ cotanhθ dθ Then x = q cosechθ

 q² + x²  = q² + q² cosech²θ =  q²(1 + cosech²θ) = q² cotanh²θ

    ———

 \/q² + x²  = | q cotanhθ |

        ———

x³ \/q² + x² = q³ cosech³θ . q cotanhθ = q⁴ cosech³θ cotanhθ

                dx                            – q cosechθ cotanhθ  dθ               – dθ                   –1               

 ò———————— . dx = ò——————————— =ò——————— =  —— òsinh²θ dθ

            ———

    x³ \/q² + x²  dx                       q⁴ cosech³θ cotanhθ             q³ cosech²θ       q³

                                        –1       1                                                                                 sinsh2θ

                                 =ò —— . — ( cosinh2θ – 1) dθ    "Note: ∫ cosinh2θ dθ =  ———— + c "

                                       q³        2                                                                                       2

                                       –1      sinsh2θ

                                 =  —— ( ———— – θ ) + c

                                      2q³             2

                                       –1      2 sinhθ cosinhθ

                                 =  —— ( ——————— – θ ) + c

                                      2q³                  2

                                       –1     

                                 =  —— ( sinhθ cosinhθ – θ ) + c

                                      2q³