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Integral of x*a*sin(x/a) dx

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01axsin(xa)dx\int\limits_{0}^{1} a x \sin{\left(\frac{x}{a} \right)}\, dx
Integral((x*a)*sin(x/a), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=axu{\left(x \right)} = a x and let dv(x)=sin(xa)\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{a} \right)}.

    Then du(x)=a\operatorname{du}{\left(x \right)} = a.

    To find v(x)v{\left(x \right)}:

    1. Let u=xau = \frac{x}{a}.

      Then let du=dxadu = \frac{dx}{a} and substitute adua du:

      asin(u)du\int a \sin{\left(u \right)}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        sin(u)du=asin(u)du\int \sin{\left(u \right)}\, du = a \int \sin{\left(u \right)}\, du

        1. The integral of sine is negative cosine:

          sin(u)du=cos(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

        So, the result is: acos(u)- a \cos{\left(u \right)}

      Now substitute uu back in:

      acos(xa)- a \cos{\left(\frac{x}{a} \right)}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    (a2cos(xa))dx=a2cos(xa)dx\int \left(- a^{2} \cos{\left(\frac{x}{a} \right)}\right)\, dx = - a^{2} \int \cos{\left(\frac{x}{a} \right)}\, dx

    1. Let u=xau = \frac{x}{a}.

      Then let du=dxadu = \frac{dx}{a} and substitute adua du:

      acos(u)du\int a \cos{\left(u \right)}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        cos(u)du=acos(u)du\int \cos{\left(u \right)}\, du = a \int \cos{\left(u \right)}\, du

        1. The integral of cosine is sine:

          cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        So, the result is: asin(u)a \sin{\left(u \right)}

      Now substitute uu back in:

      asin(xa)a \sin{\left(\frac{x}{a} \right)}

    So, the result is: a3sin(xa)- a^{3} \sin{\left(\frac{x}{a} \right)}

  3. Now simplify:

    a2(asin(xa)xcos(xa))a^{2} \left(a \sin{\left(\frac{x}{a} \right)} - x \cos{\left(\frac{x}{a} \right)}\right)

  4. Add the constant of integration:

    a2(asin(xa)xcos(xa))+constanta^{2} \left(a \sin{\left(\frac{x}{a} \right)} - x \cos{\left(\frac{x}{a} \right)}\right)+ \mathrm{constant}


The answer is:

a2(asin(xa)xcos(xa))+constanta^{2} \left(a \sin{\left(\frac{x}{a} \right)} - x \cos{\left(\frac{x}{a} \right)}\right)+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                           
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 | x*a*sin|-| dx = C + a *sin|-| - x*a *cos|-|
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axsin(xa)dx=C+a3sin(xa)a2xcos(xa)\int a x \sin{\left(\frac{x}{a} \right)}\, dx = C + a^{3} \sin{\left(\frac{x}{a} \right)} - a^{2} x \cos{\left(\frac{x}{a} \right)}
The answer [src]
  / 2    /1\        /1\\
a*|a *sin|-| - a*cos|-||
  \      \a/        \a//
a(a2sin(1a)acos(1a))a \left(a^{2} \sin{\left(\frac{1}{a} \right)} - a \cos{\left(\frac{1}{a} \right)}\right)
=
=
  / 2    /1\        /1\\
a*|a *sin|-| - a*cos|-||
  \      \a/        \a//
a(a2sin(1a)acos(1a))a \left(a^{2} \sin{\left(\frac{1}{a} \right)} - a \cos{\left(\frac{1}{a} \right)}\right)
a*(a^2*sin(1/a) - a*cos(1/a))

    Use the examples entering the upper and lower limits of integration.