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∫ x^2cos(x)dx

use integraion by parts..

let u=x^2....dv=cos(x)

du=2xdx v=sin(x)

∫ uvdv=uv-∫vdu

=x^2sin(x) - 2∫ xsin(x)dx

let u=x dv=sin(x)

du=dx v=-cos(x)

=x^2sin(x)-2[-xcos(x) - ∫ -cos(x)dx]

=x^2sin(x)-2[-xcos(x) +sin(x)]+C

=x^2sin(x)+2xcos(x)-2sin(x)+C

=x^2sin(x)-2sin(x)+2xcos(x)+C

=sin(x)(x^2-1)+2xcos(x)+C answer//

You are right of course. There's actually a neat trick to questions which require multiple integration by parts. Start as usual. Then go on inegrating one function and differentiating the other one applying alternatively plus and minus starting with plus.

Example S x^3.cos x dx

Now go on diff x^3 and inegrating cos x

=x^3(sin x)-(3x^2)(-cos x)+(6x)(-sin x)-(6)(cos x)

Cheers!

Yes, you do.

∫ x^2 cos x dx

Let u = x^2 and dv = cos x dx

then du = 2x dx and v = sin x

x^2 sin x - 2 ∫ x sin x dx

Let U = x and dV = sin x dx

then dU = dx and V = - cos x

x^2 sin x - 2x(- cos x) - 2 ∫ (- cos x dx) = x^2 sin x + 2x cos x + 2 sin x + C

Yes, you need to use integration by parts twice.

∫ x² cos(x) dx

u = x² . . . . . . . dv = cos(x) dx

du = 2x dx . . . v = sin(x)

∫ u dv = u*v - ∫ v du

∫ x² cos(x) dx = x² sin(x) - ∫ 2x sin(x) dx

∫ x² cos(x) dx = x² sin(x) - 2 ∫ x sin(x) dx

Integrate by parts again:

u = x . . . . . dv = sin(x) dx

du = dx . . . v = -cos(x)

∫ x² cos(x) dx = x² sin(x) - 2 (-x cos(x) - ∫ - cos(x) dx)

∫ x² cos(x) dx = x² sin(x) + 2x cos(x) - 2 ∫ cos(x) dx

∫ x² cos(x) dx = x² sin(x) + 2x cos(x) - 2 sin(x) + C

∫x²*cos(x) dx

Use integration by parts.

u = x²

dv = cos(x) dx

v = sin(x)

du = 2x dx

uv - ∫v du

x²*sin(x) -2*∫x*sin(x) dx

Use integration by parts again:

u = x

dv=sin(x) dx

v = -cos(x)

du = dx

∫x²*cos(x) dx = x²*sin(x) + 2x*cos(x) - 2*sin(x) + C

My wand can multitask, stimulate, de rigidity immediately, conjure up my wild and exquisite mind's eye and probably maximum magnificent of all - it may carry out a disappearing act. ;P

Your approach is correct..You can test your answer by differentiating it to get back x^2.cosx

IVAN

yes you have use to use intergation by parts again to xsin(x)