Question

Integration of (e^(5*x)+e^(3*x))/(e^x+e^ -x) give step by step explanation

Answer 1

Explanation:

`\displaystyle \\\int(e^(5x)+e^(3x))/(e^x+e^(-x))dx=\int(e^(3x)*(e^(2x)+1))/(e^x+(1)/(e^x) ) dx=\int(e^(3x)*(e^(2x)-1))/((e^(2x)+1)/(e^x) ) dx=\\\\\\=\int(e^(3x)*(e^(2x)+1)*e^x)/(e^(2x)+1) dx=\int e^(4x)dx.\\`

`\displaystyle \\Substitution:\\\\4x=u\ \ \ \ \ \ \Rightarrow\ \ \ \ \ \ x=(u)/(4) \ \ \ \ \ \ dx=(du)/(4) \\\\Hence\ \ \ \ \int e^u(du)/(4) =\int(e^u)/(4) du=(1)/(4) \int e^udu=(e^u)/(4)+C=(e^(4x))/(4) +C.`