Question

Integrate
`e^(4x)\sqrt{1+e^(2x) } dx` please show work so i can learn

Answer 1

`(\frac{(1+e^(2x)) ^{(5)/(2) } }{{5}} + \frac{(1+e^(2x) )^{(3)/(2) } }{{3}} )+C`

Explanation:

Step(i):-

Given that the function

f(x) =
`e^(4x) \sqrt{1+e^(2x) }`

Now integrating on both sides, we get

`\int\limits{f(x)} \, dx = \int\limits{e^(4x) \sqrt{1+e^(2x) } dx`

=
`\int\limits{e^(2x) e^(2x) \sqrt{1+e^(2x) } dx`

Step(ii):-

Let
`1 + e^(2x) = t`

`e^(2x) = t -1`

`2e^(2x)dx = d t`

`e^(2x)dx = (1)/(2) d t`

=
`\int\limits{( \sqrt{1+e^(2x) }) e^(2x) e^(2x) dx`

=
`\int\limits {√(t)(t-1)(1)/(2) dt }`

=
`(1)/(2) \int\limits {√(t) (t) -√(t) ) dt }`

=
`(1)/(2) \int\limits {(t^{(1)/(2) } t^(1) +t^{(1)/(2) } ) } \, dx`

`= (1)/(2) \int\limits {(t^{(3)/(2) } +t^{(1)/(2) } ) } \, dx`

=
`(1)/(2) (\frac{t^{(3)/(2) +1} }{(3)/(2)+1 } + \frac{t^{(1)/(2) +1} }{(1)/(2)+1 } )+C`

=
`(1)/(2) (\frac{t^{(3)/(2) +1} }{(5)/(2) } + \frac{t^{(1)/(2) +1} }{(3)/(2) } )+C`

=
`(1)/(2) (\frac{t^{(5)/(2) } }{(5)/(2) } + \frac{t^{(3)/(2) } }{(3)/(2) } )+C`

=
`(\frac{(1+e^(2x)) ^{(5)/(2) } }{{5}} + \frac{(1+e^(2x) )^{(3)/(2) } }{{3}} )+C`

Final answer:-

=
`(\frac{(1+e^(2x)) ^{(5)/(2) } }{{5}} + \frac{(1+e^(2x) )^{(3)/(2) } }{{3}} )+C`