now-try-this-one-dx-x-1-2-x-1-3-x-1-6-

Question Number 69637 by MJS last updated on 26/Sep/19

\dots now try this one :

\int \frac{d x}{x^{1 / 2} - x^{1 / 3} - x^{1 / 6}} =

Answered by Kunal12588 last updated on 26/Sep/19

t = x^{1 / 6}

\Rightarrow d t = \frac{1}{6} x^{- 5 / 6} d x \Rightarrow d x = 6 x^{5 / 6} d t = 6 t^{5} d t

\int \frac{6 t^{5}}{t^{3} - t^{2} - t} d t = 6 \int \frac{t^{4}}{t^{2} - t - 1} d t

t^{4} = (t^{2} - t - 1) (t^{2} + t + 2) + 3 t + 2

I = 6 \int (t^{2} + t + 2) d t + 6 \int \frac{3 t + 2}{t^{2} - t - 1} d t

I_{1} = \int \frac{3 t - 2}{t^{2} - t + 1} d t

3 t - 2 = A \frac{d}{d t} (t^{2} - t + 1) + B

\Rightarrow A = \frac{3}{2}, B = \frac{7}{2}

I_{1} = \frac{3}{2} \int \frac{d (t^{2} - t + 1)}{t^{2} - t + 1} + \frac{7}{2} \int \frac{d t}{t^{2} - t + 1}

I_{1} = \frac{3}{2} l o g ∣ t^{2} - t + 1 ∣ + \frac{7}{2} \int \frac{d t}{{(t - \frac{1}{2})}^{2} + \frac{3}{4}}

I_{1} = \frac{3}{2} l o g ∣ t^{2} - t + 1 ∣ + \frac{7}{2} \times \frac{1}{2 \sqrt{3}} {t a n}^{- 1} (\frac{t - \frac{1}{2}}{\frac{\sqrt{3}}{2}}) + c

I_{1} = \frac{3}{2} l o g ∣ t^{2} - t + 1 ∣ + \frac{7}{4 \sqrt{3}} {t a n}^{- 1} (\frac{2 t - 1}{\sqrt{3}}) + c

I = 2 t^{3} + 3 t^{2} + 12 t + 9 l o g (t^{2} - t + 1) + \frac{7 \sqrt{3}}{2} {t a n}^{- 1} (\frac{2 t - 1}{\sqrt{3}}) + c

I = 2 x^{1 / 2} + 3 x^{1 / 3} + 12 x^{1 / 6} + 9 l o g (x^{1 / 3} - x^{1 / 6} + 1) + 7 \sqrt{3} {t a n}^{- 1} (\frac{2 \sqrt{x^{1 / 3} - x^{1 / 6} + 1} - \sqrt{3}}{2 x^{1 / 6} - 1}) + c