Find the integral of f(x) such that f(x) = x*sinx

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

f(x) = x*sinx

We need to determine the indefinite integral of f(x)

we note that the function is aproduct of two terms.

==> intg f(x) = intg x*sinx dx

Then we will apply the rule:

Let f(x) = u*dv    such that:

u= x    ==>    du = dx

dv= sinx dx  ==>  v = intg sinx dx = -cosx

==> we know that:

intg f(x) = u*v - intg v*du

             = x*(-cosx) - intg (-cosx) dx

               = -xcosx + sinx

==> intg f(x) = -xcosx + sinx

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

To find integral of f(x)  such that f(x) = x*sinx.

Since f(x) = xsinx.

Intf(x) dx = Int x*sinx dx.

We know that  Int u(x) v(x) = u(x) int v(x) dx - Int {u'(x) int v(x) dx} dx

Intf(x) dx =  x Int (sinx dx) - Int {(x)' int (sinx) dx}dx.

Int f(x) dx = x(- cosx) - Int {1* (-cosx)}dx.

Int f(x) dx = -xcosx +sinx.

Therefore Integral of f(x) dx =  Integral (xsinx) dx =  sinx-xcosx +C, where C is the contant of integration.

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