1-find-f-x-0-1-ln-1-xt-3-dt-with-x-lt-1-2-calculate-0-1-ln-2-t-3-dt-

Question Number 45635 by maxmathsup by imad last updated on 14/Oct/18

1) f i n d f (x) = \int_{0}^{1} l n (1 + {x t}^{3}) d t w i t h ∣ x ∣< 1

2) c a l c u l a t e \int_{0}^{1} l n (2 + t^{3}) d t .

Commented by maxmathsup by imad last updated on 17/Oct/18

1) c h a n g e m e n t^{3} \sqrt{x} t = u g i v e

f (x) = \int_{0}^{3_{\sqrt{x}}} l n (1 + u^{3}) \frac{d u}{(^{3} \sqrt{x})} = \frac{1}{(^{3} \sqrt{x})} \int_{0}^{3_{\sqrt{x}}} l n (1 + u^{3}) d u b u t w e h a v e p r o v e d t h a t

\int l n (1 + x^{3}) d x = x l n (1 + x^{3}) - 3 x + l n ∣ x + 1 ∣ - \frac{1}{2} l n (x^{2} - x + 1) + \sqrt{3} a r c t a n (\frac{2 x - 1}{\sqrt{3}})

f (x) = \frac{1}{(^{3} \sqrt{x})} {[u l n (1 + u^{3}) - 3 u + l n ∣ u + 1 ∣ - \frac{1}{2} l n (u^{2} - u + 1) + \sqrt{3} a r c t a n (\frac{2 x - 1}{\sqrt{3}})]}_{0}^{3_{\sqrt{x}}}

= \frac{1}{(^{3} \sqrt{x})} {^{3} \sqrt{x} l n (1 + x) - 3^{} (^{3} \sqrt{x}) + l n ∣ 1 +^{3} \sqrt{x} ∣ - \frac{1}{2} l n {u^{\frac{2}{3}} -^{3} \sqrt{x} + 1}

+ \sqrt{3} a r c t a n (\frac{2 (^{3} \sqrt{x}) - 1}{\sqrt{3}}) + \frac{π \sqrt{3}}{6}} .

Commented by maxmathsup by imad last updated on 17/Oct/18

l e t A = \int_{0}^{1} l n (2 + t^{3}) d t \Rightarrow A = \int_{0}^{1} l n {2 (1 + \frac{1}{2} t^{3})} d t

= l n (2) + \int_{0}^{1} l n (1 + \frac{1}{2} t^{3}) d t

= l n (2) + f (\frac{1}{2}) = l n (2) +^{3} \sqrt{2} {^{3} \sqrt{2} l n (\frac{3}{2}) - 3 (^{3} \sqrt{2}) + l n ∣ 1 +^{3} \sqrt{2} ∣

- \frac{1}{2} l n {2^{- \frac{2}{3}} - \frac{1}{(^{3} \sqrt{x})} + 1} + \sqrt{3} a r c t a n (\frac{\frac{2}{(^{3} \sqrt{2})} - 1}{\sqrt{3}}) + \frac{π \sqrt{3}}{6}} .