0-x-t-2-a-2t-2-dt-

Question Number 166141 by Lambert last updated on 13/Feb/22

\int_{0}^{x} \frac{t^{2}}{\sqrt{a + 2 t^{2}}} dt

Answered by MJS_new last updated on 14/Feb/22

\int \frac{t^{2}}{\sqrt{a + 2 t^{2}}} d t =

[u = \frac{\sqrt{2}}{\sqrt{a}} (t + \sqrt{a + 2 t^{2}}) \to d t = \frac{\sqrt{a + 2 t^{2}}}{\sqrt{2} u} d u]

= \frac{a \sqrt{2}}{16} \int \frac{{(u^{2} - 1)}^{2}}{u^{3}} d u = \frac{a \sqrt{2}}{16} \int (u - \frac{2}{u} + \frac{1}{u^{3}}) d u =

= \frac{a \sqrt{2}}{16} (\frac{u^{2}}{2} - 2 \ln u - \frac{1}{2 u^{2}}) = \dots

= \frac{t \sqrt{a + 2 t^{2}}}{4} - \frac{a \sqrt{2}}{8} \ln (\sqrt{2} t + \sqrt{a + 2 t^{2}}) + C

the rest is easy

Commented by cortano1 last updated on 14/Feb/22

u = \frac{\sqrt{2}}{\sqrt{a}} t ? sir

Commented by MJS_new last updated on 14/Feb/22

typo = instead of +; {it}^{'} s now corrected

Answered by Mathspace last updated on 14/Feb/22

f (a) = \frac{1}{\sqrt{a}} \int_{0}^{x} \frac{t^{2}}{\sqrt{1 + \frac{2 t^{2}}{a}}} d t

= \frac{a}{2} \times \frac{1}{\sqrt{a}} \int_{0}^{x} \frac{\frac{2}{a} t^{2}}{\sqrt{1 + \frac{2 t^{2}}{a}}} d t

= \frac{\sqrt{a}}{2} \int_{0}^{x} \sqrt{1 + \frac{2 t^{2}}{a}} d t - \frac{\sqrt{a}}{2} \int_{0}^{x} \frac{d t}{\sqrt{1 + \frac{2 t^{2}}{a}}}

\frac{\sqrt{2} t}{\sqrt{a}} = s h u \Rightarrow u = a r g s h (\frac{\sqrt{2} t}{\sqrt{a}})

\int_{0}^{x} \sqrt{1 + \frac{2 t^{2}}{a}} d t = \int_{0}^{a r g s h (\frac{\sqrt{2} x}{\sqrt{a}})} c h u . \frac{\sqrt{a}}{\sqrt{2}} c h u d u

= \frac{\sqrt{a}}{\sqrt{2}} \int_{0}^{l n (\frac{\sqrt{2} x}{\sqrt{a}} + \sqrt{1 + \frac{2 x^{2}}{a}})} \frac{1 + c h (2 u)}{2} d u

= \frac{\sqrt{a}}{2} l n (\dots .) + \frac{\sqrt{a}}{4 \sqrt{2}} {[s h (2 u)]}_{0}^{l n (\dots)}

= \frac{\sqrt{a}}{2} l n (\dots) + \frac{\sqrt{a}}{4 \sqrt{2}} {[\frac{e^{2 u} - e^{- 2 u}}{2}]}_{0}^{l n (\dots)}

= \frac{\sqrt{a}}{2} l n (\frac{\sqrt{2} x}{\sqrt{a}} + \sqrt{1 + \frac{2 x^{2}}{a}})

\frac{\sqrt{a}}{8 \sqrt{2}} {{(\frac{\sqrt{2} x}{\sqrt{a}} + \sqrt{1 + \frac{2 x^{2}}{a}})}^{2} - {(\frac{\sqrt{2} x}{\sqrt{a}} + \sqrt{1 + \frac{2 x^{2}}{a}})}^{- 2}}

s a m e w a y f o r o t h e r i n t e g r a l \dots