calculate-0-1-x-3-2-x-2-3-dx-

Question Number 60050 by maxmathsup by imad last updated on 17/May/19

c a l c u l a t e \int_{0}^{1} (x^{3} - 2) \sqrt{x^{2} + 3} d x

Commented by maxmathsup by imad last updated on 18/May/19

l e t A = \int_{0}^{1} (x^{3} - 2) \sqrt{x^{2} + 3} d x \Rightarrow A = \int_{0}^{1} x^{3} \sqrt{x^{2} + 3} d x - 2 \int_{0}^{1} \sqrt{x^{2} + 3} d x = H - K

c h a n g e m e n t \sqrt{x^{2} + 3} = t g i v e x^{2} + 3 = t^{2} \Rightarrow 2 x d x = 2 t d t \Rightarrow

A = \int_{\sqrt{3}}^{2} (t^{2} - 3) t t d t = \int_{\sqrt{3}}^{2} t^{2} (t^{2} - 3) d t = \int_{\sqrt{3}}^{2} (t^{4} - 3 t^{2}) d t

= {[\frac{1}{5} t^{5} - t^{3}]}_{\sqrt{3}}^{2} = \frac{32}{5} - 8 - \frac{1}{5} {(\sqrt{3})}^{5} + {(\sqrt{3})}^{3}

\int_{0}^{1} \sqrt{x^{2} + 3} d x =_{x = \sqrt{3} s h (t)} \int_{0}^{a r g s h (\frac{1}{\sqrt{3}})} \sqrt{3} c h (t) \sqrt{3} c h (t) d t

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

= \frac{3}{2} l n (\sqrt{3}) + \frac{3}{4} {[s h (2 t)]}_{0}^{l n (\sqrt{3})} = \frac{3}{2} l n (\sqrt{3}) + \frac{3}{8} {[e^{2 t} - e^{- 2 t}]}_{0}^{l n (\sqrt{3})}

= \frac{3}{4} l n (3) + \frac{3}{8} {{(\sqrt{3})}^{2} - \frac{1}{{(\sqrt{3})}^{2}}} = \frac{3}{4} l n (3) + \frac{3}{8} {3 - \frac{1}{3}}

= \frac{3}{4} l n (3) + 1 s o t h e v a l u e o f A i s d e t e r m i n e d .

Answered by tanmay last updated on 17/May/19

\int x^{3} \sqrt{x^{2} + 3} d x - 2 \int \sqrt{x^{2} + 3} d x

I_{1} = \int x^{3} \sqrt{x^{2} + 3} d x

t^{2} = x^{2} + 3

2 t d t = 2 x d x

\int x^{2} \times x \sqrt{x^{2} + 3} d x

\int (t^{2} - 3) \times t \times t d t

\int (t^{4} - 3 t^{2}) d t

= \frac{t^{5}}{5} - t^{3} + c

= \frac{{(x^{2} + 3)}^{\frac{5}{2}}}{5} - {(x^{2} + 3)}^{\frac{3}{2}} + c

I_{2} = \int \sqrt{x^{2} + 3} d x \to u s e f o r m u l a

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

I = I_{1} - 2 I_{2}

I =∣ \frac{{(x^{2} + 3)}^{\frac{5}{2}}}{5} - {(x^{2} + 3)}^{\frac{3}{2}} - x \sqrt{x^{2} + 3} - 3 l n (x + \sqrt{x^{2} + 3}) ∣_{0}^{1}

= \frac{1}{5} {{(4)}^{\frac{5}{2}} - {(3)}^{\frac{5}{2}}} - {{(4)}^{\frac{3}{2}} - 3^{\frac{3}{2}}} - {\sqrt{4} - 0} - 3 l n (\frac{3}{\sqrt{3}})

= \frac{32}{5} - \frac{9}{5} \sqrt{3} - 8 + 3 \sqrt{3} - 2 - \frac{3}{2} l n 3

= \frac{32}{5} - 10 + 3 \sqrt{3} (1 - \frac{3}{5}) - \frac{3}{2} l n 3

= \frac{- 18}{5} + \frac{6}{5} \sqrt{3} - \frac{3}{2} l n 3

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