∫_0 ^1 ∫_0 ^1 cos(max(x^3 ,y^(3/2) ))dxdy=A old Quation By mr,univers x^3 =t,y^(3/2) =s A=(2/9)∫_0 ^1 ∫_0 ^1 cos(max(t,s))t^(−(2/3)) s^(−(1/3)) dtds ∫_0 ^1 ∫_0 ^1 cos(max(t,s))t^(−(2/3)) s^(−(1/3)) dtds=∫_0 ^1 t^(−(2/3)) ∫_t ^1 cos(s)s^(−(1/3)) dsdt_(=C) +∫_0 ^1 s^(−(1/3)) ∫_s ^1 cos(t)t^(−(2/3)) dtds_(=B) c=∫_0 ^1 t^(−(2/3)) ∫_t ^1 Σ_(n≥0) (((−1)^n )/(2n!)).s^(2n−(1/3)) dsdt=Σ_(n≥0) (((−1)^n )/(2n!))∫_0 ^1 t^(−(2/3)) ∫_t ^1 s^(2n−(1/3)) dsdt =Σ_(n≥0) (((−1)^n )/((2n)!))∫_0 ^1 ((1−t^(2n+(2/3)) )/(2n+(2/3))).t^(−(2/3)) dt =Σ_(n≥0) (((−1)^n )/((2n)!(2n+(2/3)))).(3−(1/(2n+1)))=3Σ_(n≥0) (((−1)^n )/((2n+1)!))=3sin(1) B=∫_0 ^1 s^(−(1/3)) ∫_s ^1 Σ_(n≥0) (((−1)^n )/((2n)!))t^(2n−(2/3)) dtds =Σ(((−1)^n )/((2n)!))∫_0 ^1 s^(−(1/3)) .(((1−s^(2n+(1/3)) )/(2n+(1/3))))ds =Σ_(n≥0) (((−1)^n )/((2n)!(2n+(1/3))))((3/2)−(1/(2n+1))) =(3/2)Σ_(n≥0) (((−1)^n )/((2n+1)!))=(3/2)sin(1) A=(2/9)(c+B)=(2/9)((3/2)sin(1)+3sin(1)) =sin(1) ∫_0 ^1 ∫


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 199638 by witcher3 last updated on 06/Nov/23

$\int_{0}^{1} \int_{0}^{1} \cos (\max (x^{3}, y^{\frac{3}{2}})) dxdy = A$ $old Quation By mr, univers$ $x^{3} = t, y^{\frac{3}{2}} = s$ $A = \frac{2}{9} \int_{0}^{1} \int_{0}^{1} \cos (\max (t, s)) t^{- \frac{2}{3}} s^{- \frac{1}{3}} dtds$ $\int_{0}^{1} \int_{0}^{1} \cos (\max (t, s)) t^{- \frac{2}{3}} s^{- \frac{1}{3}} dtds = \int_{0}^{1} t^{- \frac{2}{3}} \int_{t}^{1} \cos (s) s^{- \frac{1}{3}} {dsdt}_{= C}$ $+ \int_{0}^{1} s^{- \frac{1}{3}} \int_{s}^{1} \cos (t) t^{- \frac{2}{3}} {dtds}_{= B}$ $c = \int_{0}^{1} t^{- \frac{2}{3}} \int_{t}^{1} \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{2 n!} . s^{2 n - \frac{1}{3}} dsdt = \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{2 n!} \int_{0}^{1} t^{- \frac{2}{3}} \int_{t}^{1} s^{2 n - \frac{1}{3}} dsdt$ $= \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n)!} \int_{0}^{1} \frac{1 - t^{2 n + \frac{2}{3}}}{2 n + \frac{2}{3}} . t^{- \frac{2}{3}} dt$ $= \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n)! (2 n + \frac{2}{3})} . (3 - \frac{1}{2 n + 1}) = 3 \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n + 1)!} = 3 \sin (1)$ $B = \int_{0}^{1} s^{- \frac{1}{3}} \int_{s}^{1} \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n)!} t^{2 n - \frac{2}{3}} dtds$ $= Σ \frac{{(- 1)}^{n}}{(2 n)!} \int_{0}^{1} s^{- \frac{1}{3}} . (\frac{1 - s^{2 n + \frac{1}{3}}}{2 n + \frac{1}{3}}) ds$ $= \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n)! (2 n + \frac{1}{3})} (\frac{3}{2} - \frac{1}{2 n + 1})$ $= \frac{3}{2} \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n + 1)!} = \frac{3}{2} \sin (1)$ $A = \frac{2}{9} (c + B) = \frac{2}{9} (\frac{3}{2} \sin (1) + 3 \sin (1))$ $= \sin (1)$ $\int_{0}^{1} \int_{0}^{1} \cos (Max (x^{3}, y^{\frac{3}{2}})) dxdy = \sin (1)$
Commented by universe last updated on 06/Nov/23

$t h a n k s s i r$
Commented by witcher3 last updated on 06/Nov/23

$withe Pleasur$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum