Integrate the function 1 x^ 1 2 +x^ 1 3 [Hint: 1 x^ 1 2 +x^ 1 3 =1 x^ 1 3 (1+x^ 1 6 ) Put x = t^6]

Given: 1 x^ 1 2 +x^ 1 3 or we can write it as 1 x^ 1 2 +x^ 1 3 =1 x^ 1 3 (1+x^ 1 6 ) Let x = t^ 6 dx = 6t^5dt 1 x^ 1 3 (1+x^ 1 6 ) d x= 6 t^ 5 t^ 2 (1+t) d t = 6. t^ 3 (1+t) d t After division we get, 1 x^ 1 2 +x^ 1 3 = [ (t^ 2 -t+1 )-1 (1+t) ] d t = 6 . t^ 2 . dt- t . dt+ 1 . dt- [1 (1+t) ] dt = 6 [ ( t^ 3 3 )- ( t^ 2 2 )+t- (1+t) ] = 6 [ ( (x^ 1 6 )^ 3 3 )- ( (x^ 1 6 )^ 2 2 )+ (x^ 1 6 )- (1+ (x^ 1 6 ) ) ]+c = [ (2 x^ 1 2 )- (3 x^ 1 3 )+6 x^ 1 6 -6 (1+x^ 1 6 ) ]+C = 2 x-3 x^ 1 3 +6 x^ 1 6 -6 (1+x^ 1 6 )+c

Integrate the function 1 x^ 1 2 +x^ 1 3 [Hint: 1 x^ 1 2 +x^ 1 3 =…

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Question

Integrate the function $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)}$ Put $x = t^6]$

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Answer

Given: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ or we can write it as $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)} $
Let $x = t^{6 } $
$\Rightarrow dx = 6t^5dt$
$\Rightarrow \int \frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)} \cdot d x=\int \frac{6 t^{5}}{t^{2}(1+t)} \cdot d t $
$= \text { 6. } \int \frac{t^{3}}{(1+t)} \cdot d t $
After division we get,
$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}} = \int\left[\left(t^{2}-t+1\right)-\frac{1}{(1+t)}\right] \cdot d t $
$= 6 .\left\{\int \mathrm{t}^{2} . \mathrm{dt}-\int \mathrm{t} . \mathrm{dt}+\int 1 . \mathrm{dt}-\int\left[\frac{1}{(1+t)}\right] \cdot \mathrm{dt}\right\} $
$= 6\left[\left(\frac{t^{3}}{3}\right)-\left(\frac{t^{2}}{2}\right)+t-\log (1+t)\right] $
$= 6\left[\left(\frac{\left(x^{\frac{1}{6}}\right)^{3}}{3}\right)-\left(\frac{\left(x^{\frac{1}{6}}\right)^{2}}{2}\right)+\left(x^{\frac{1}{6}}\right)-\log \left(1+\left(x^{\frac{1}{6}}\right)\right)\right]+c $
$= \left[\left(2 x^{\frac{1}{2}}\right)-\left(3 x^{\frac{1}{3}}\right)+6 \cdot x^{\frac{1}{6}}-6 \cdot \log \left(1+x^{\frac{1}{6}}\right)\right]+C $
$= 2 \sqrt{x}-3 x^{\frac{1}{3}}+6 x^{\frac{1}{6}}-6 \log \left(1+x^{\frac{1}{6}}\right)+c $

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