x-5-2-1-x-3-2-dx-

Question Number 33619 by NECx last updated on 20/Apr/18

\int x^{5 / 2} {(1 - x)}^{3 / 2} d x

Commented by abdo imad last updated on 20/Apr/18

l e t p u t I = \int x^{\frac{5}{2}} {(1 - x)}^{\frac{3}{2}} d x . c h . x = {s i n}^{2} t g i v e

I = \int {s i n}^{5} x {c o s}^{3} x (2 s i n t c o s t) d t

= 2 \int {c o s}^{4} x {s i n}^{6} x d x

= 2 \int {(c o s x s i n x)}^{4} {s i n}^{2} x d x

= \frac{1}{8} \int {(s i n (2 x))}^{4} \frac{1 - c o s (2 x)}{2} d x

= \frac{1}{16} \int {(\frac{1 - c o s (4 x)}{2})}^{2} (1 - c o s (2 x)) d x

= \frac{1}{64} \int (1 - 2 c o s (4 x) + {c o s}^{2} (4 x)) (1 - c o s (2 x)) d x

= \frac{1}{64} \int (1 - 2 c o s (4 x) + \frac{1 + c o s (8 x)}{2}) (1 - c o s (2 x)) d x

= \frac{1}{128} \int (2 - 4 c o s (4 x) + 1 + c o s (8 x)) (1 - c o s (2 x)) d x

= \dots . . u n t i l w e f i n d t h e v a l u e \dots .

Commented by abdo imad last updated on 20/Apr/18

t h i s m e t h o d i s u s e f u l i f - 1 ⩽ x ⩽ 1 .

Commented by NECx last updated on 21/Apr/18

c a n t i t b e c o m p l e t e d ?

Answered by MJS last updated on 21/Apr/18

u = \sqrt{x}

d x = 2 \sqrt{x} d u

2 \int u^{6} {(1 - u^{2})}^{\frac{3}{2}} d u

u = \sin v

d u = \cos v d v

2 \int \cos v \sin^{6} v {(1 - \sin^{2} v)}^{\frac{3}{2}} d v =

= 2 \int \cos^{4} v \sin^{6} v d v = 2 \int {(1 - \sin^{2} v)}^{2} \sin^{6} v d v =

= 2 (\int \sin^{10} v d v - 2 \int \sin^{8} v d v + \int \sin^{6} v d v)

now we have to use

\int \sin^{n} α d α = \frac{n - 1}{n} \int \sin^{n - 2} α d α - \frac{\cos α \sin^{n - 1} α}{n}

until {we}^{'} re done and get

\frac{3 v}{128} + \cos v (- \frac{1}{5} \sin^{9} v + \frac{11}{40} \sin^{7} v - \frac{1}{80} \sin^{5} v - \frac{1}{64} \sin^{3} v - \frac{3}{128} \sin v)

now we go back \dots

v = \arcsin u

\sin v = u

\cos v = \sqrt{1 - u^{2}}

u = \sqrt{x}

\int x^{5 / 2} {(1 - x)}^{3 / 2} d x =

= \frac{3}{128} \arcsin \sqrt{x} - \frac{\sqrt{x (1 - x)}}{640} (128 x^{4} - 176 x^{3} + 8 x^{2} + 10 x + 15) + C

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