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0-1-4x-3-d-2-dx-2-1-x-2-5-dx-




Question Number 4957 by ankit chakravarti last updated on 26/Mar/16
∫_0 ^1 4x^3 {(d^2 /dx^2 )(1−x^2 )^5 }dx=?
014x3{d2dx2(1x2)5}dx=?
Commented by ankit chakravarti last updated on 26/Mar/16
please solve.....
pleasesolve..
Answered by Yozzii last updated on 26/Mar/16
Integrating by parts we obtain  I=∫_0 ^1 4x^3 {(d^2 /dx^2 )(1−x^2 )^5 }dx  I=4x^3 (d/dx)(1−x^2 )^5 ∣_0 ^1 −∫_0 ^1 12x^2 ((d/dx)(1−x^2 )^5 )dx  I=0−12([x^2 (1−x^2 )^5 ]_0 ^1 −∫_0 ^1 2x(1−x^2 )^5 dx)  I=(−12)∫_0 ^1 (−2x)(1−x^2 )^5 dx  Let u=1−x^2 ⇒du=−2xdx  ⇒I=−12∫_1 ^0 u^5 du=12∫_0 ^1 u^5 du=((12)/6)u^6 ∣_0 ^1 =2.
IntegratingbypartsweobtainI=014x3{d2dx2(1x2)5}dxI=4x3ddx(1x2)5010112x2(ddx(1x2)5)dxI=012([x2(1x2)5]01012x(1x2)5dx)I=(12)01(2x)(1x2)5dxLetu=1x2du=2xdxI=1210u5du=1201u5du=126u601=2.

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