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Solving integrals/ x*a*sin(x/a)

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

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The solution

You have entered [src] 
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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:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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=a∫sin⁡(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=−a2∫cos⁡(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=a∫cos⁡(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\           3    /x\      2    /x\
 | 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)}
Simplify
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.

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