find-x-2-x-3-dx-

Question Number 56700 by maxmathsup by imad last updated on 21/Mar/19

f i n d \int \sqrt{x - 2 \sqrt{x} + 3} d x

Commented by kaivan.ahmadi last updated on 22/Mar/19

Commented by maxmathsup by imad last updated on 23/Mar/19

I = \int \sqrt{{(\sqrt{x} - 1)}^{2} + 2} d x c h a n g e m e n t \sqrt{x} - 1 = \sqrt{2} s h (t) g i v e \sqrt{x} = 1 + \sqrt{2} s h t \Rightarrow

x = {(1 + \sqrt{2} s h (t))}^{2} \Rightarrow

I = \int \sqrt{2 {s h}^{2} t + 2} (2 (1 + \sqrt{2} s h (t)) \sqrt{2} c h t d t

= 4 \int {c h}^{2} (t) (1 + \sqrt{2} s h (t)) d t = 4 \int {c h}^{2} t d t + 4 \sqrt{2} \int s h (t) {c h}^{2} t d t

= 4 \int \frac{1 + c h (2 t)}{2} d t + 4 \sqrt{2} \frac{1}{3} {c h}^{3} (t) = 2 t + 2 \int c h (2 t) d t + \frac{4 \sqrt{2}}{3} {c h}^{3} (t)

= 2 t + s h (2 t) + \frac{4 \sqrt{2}}{3} {c h}^{3} (t) b u t t = a r g s h (\frac{\sqrt{x} - 1}{2}) = l n (\frac{\sqrt{x} - 1}{2} + \sqrt{1 + {(\frac{\sqrt{x} - 1}{2})}^{2}})

I = 2 a r g s h (\frac{\sqrt{x} - 1}{2}) + s h (2 a r g s h (\frac{\sqrt{x} - 1}{2})) + \frac{4 \sqrt{2}}{3} {c h}^{3} (a r g s h (\frac{\sqrt{x} - 1}{2})) + c

Answered by kaivan.ahmadi last updated on 22/Mar/19

\int \sqrt{{(\sqrt{x} - 1)}^{2} + 2} d x =

t = \sqrt{x} - 1 \Rightarrow d t = \frac{d x}{2 \sqrt{x}} = \frac{d x}{2 t + 2} \Rightarrow d x = (2 t + 2) d t

\int \sqrt{t^{2} + 2} (2 t + 2) d t = \int 2 t \sqrt{t^{2} + 2} d t + \int 2 \sqrt{t^{2} + 2} d t = I + J

s o l v e I

u = t^{2} + 2 \Rightarrow d u = 2 t d t

\int \sqrt{u} d u = \frac{2}{3} u^{\frac{3}{2}} + C_{1} = \frac{2}{3} {(t^{2} + 2)}^{\frac{3}{2}} + C_{1} =

{({(\sqrt{x} - 1)}^{2} + 2)}^{\frac{3}{2}} + C_{1}

s o l v e J

t = \sqrt{2} t g θ \Rightarrow d t = \sqrt{2} (1 + {t g}^{2} θ) d θ

a n d s o t g θ = \frac{t}{\sqrt{2}} \Rightarrow s e c θ = \sqrt{1 + \frac{t^{2}}{2}}

\int 2 \sqrt{2 (1 + {t g}^{2} θ)} . \sqrt{2} (1 + {t g}^{2} θ) d θ =

\int 4 (1 + {t g}^{2} θ) \sqrt{1 + {t g}^{2} θ} d θ = \int 4 \frac{d θ}{{c o s}^{3} θ} =

4 \int {s e c}^{3} θ d θ = 4 (\frac{1}{2} s e c θ t g θ + \frac{1}{2} (s e c θ + t g θ)) + C_{2} =

2 s e c θ t g θ + 2 (s e c θ + t g θ) + C_{2} =

2 \sqrt{1 + \frac{t^{2}}{2}} . \frac{t}{\sqrt{2}} + 2 (\sqrt{1 + \frac{t^{2}}{2}} + \frac{t}{\sqrt{2}}) + C_{2}