Question

Find the indefinite integral of
`\int\limits {(5)/(x^(1)/(2)+x^(3)/(2)) \, dx`

I have been able to simplify it to
`\int\limits {(5√(x))/(x^3+x)} \, dx` but that is confusing,

I then did u-subsitution where
`u=√(x)` to obtain
`\int\limits {(5u)/(u^6+u^2)} \, dx` which simplified to
`\int\limits {(5)/(u^5+u)} \, dx`, a much nicer looking integrand
however, I am still stuck

ples help
show all work or be reported

Answer 1

`\displaystyle10 \tan^(-1)( \sqrt {x}^( ) ) + \rm C`

Explanation:

we would like to integrate the following integration:

`\displaystyle \int \frac{5}{ {x}^{ (1)/(2) } + {x}^{ (3)/(2) } } dx`

in order to do so we can consider using u-substitution

let our

`\displaystyle u = {x}^{ (1)/(2) } \quad \text{and} \quad du = \frac{ {x}^{ - (1)/(2) } }{2}`

to apply substitution we need a little bit arrangement

multiply both Integral and integrand by 2 and ½

`\displaystyle 2\int (1)/(2) \cdot\frac{5}{ {x}^{ (1)/(2) } + {x}^{ (3)/(2) } } dx`

factor out
`x^{(1)/(2)}`:

`\displaystyle 2\int \frac{1}{2 {x}^{ (1)/(2) } } \cdot\frac{5}{ (1+ {x}^( ) )} dx`

recall law of exponent:

`\displaystyle 2\int \frac{ {x}^{ - (1)/(2) } }{2 } \cdot\frac{5}{ (1+ {x}^( ) )} dx`

apply substitution:

`\displaystyle 2\int \frac{5}{ 1+ {x}^( ) } du`

rewrite x as
`\big(x^{(1)/(2)}\big)^(2)`:

`\displaystyle 2\int \frac{5}{ 1+ ( {x ^{ (1)/(2) } })^( 2 ) } du`

substitute:

`\displaystyle 2\int (5)/( 1+ ( u)^( 2 ) ) du`

recall integration rule of inverse trig:

`\displaystyle 2 * 5 \tan^(-1)(u)`

simplify multiplication:

`\displaystyle10 \tan^(-1)(u)`

substitute back:

`\displaystyle10 \tan^(-1)( {x}^{ (1)/(2) } )`

simplify if needed:

`\displaystyle10 \tan^(-1)( √(x) )`

finally we of course have to add constant of integration:

`\displaystyle10 \tan^(-1)( \sqrt {x}^( ) ) + \rm C`

and we are done!

Answer 2

The easiest way to calculate this integral is substitution.


`$\int(5)/(x^(1)/(2)+x^(3)/(2))\,dx=5\int(1)/(x^(1)/(2)+(x^(1)/(2))^3)\,dx=5\int(1)/(√(x)+(√(x))^3)\,dx=(\star)`

Now we can substitute
`u=√(x)` and then:


`du=(1)/(2√(x))\,dx\qquad\implies\qquad dx=2√(x)\,du=2u\,du`

So:


`$(\star)=5\int(1)/(√(x)+(√(x))^3)\,dx=5\int(2u)/(u+u^3)\,du=10\int(1)/(1+u^2)\,dx=`


`=10\arctan(u)+C=\boxed{10\arctan(√(x))+C}`