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Question

Mathematics Question on integral

Integrate the function: e3log x(x4+1)−1e^{3log\ x}(x^4+1)^{-1}

Answer

e3log x(x4+1)−1e^{3log\ x}(x^4+1)^{-1} =elog x3(x4+1)−1e^{log\ x^3}(x^4+1)^{-1} = x3(x4+1)\frac {x^3}{(x^4+1)}

Let x4+1=t⇒4x3dx=dtLet \ x^4+1=t⇒4x^3dx=dt

⇒∫$$e^{3log\ x}(x^4+1)^{-1}\ dx =∫$$\frac {x^3}{(x^4+1)}\ dx

                                     =$\frac 14∫\frac {dt}{t}$

                                     =$\frac 14log\ |t|+C$

                                     =$\frac 14log|x^4+1|+C$

                                     =$\frac 14log(x^4+1)+C$