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Solving integrals/ x(cosx-1/2)

Integral of x(cosx-1/2) dx

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The solution

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∫−π3π3x(cos⁡(x)−12) dx\int\limits_{- \frac{\pi}{3}}^{\frac{\pi}{3}} x \left(\cos{\left(x \right)} - \frac{1}{2}\right)\, dx 
Integral(x*(cos(x) - 1/2), (x, -pi/3, pi/3))
Detail solution

  1. Rewrite the integrand:

    x(cos⁡(x)−12)=xcos⁡(x)−x2x \left(\cos{\left(x \right)} - \frac{1}{2}\right) = x \cos{\left(x \right)} - \frac{x}{2}

  2. Integrate term-by-term:

    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)=xu{\left(x \right)} = x and let dv⁡(x)=cos⁡(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

      Then du⁡(x)=1\operatorname{du}{\left(x \right)} = 1.

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

      1. The integral of cosine is sine:

        ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

      Now evaluate the sub-integral.

    2. The integral of sine is negative cosine:

      ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

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

      ∫(−x2) dx=−∫x dx2\int \left(- \frac{x}{2}\right)\, dx = - \frac{\int x\, dx}{2}

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: −x24- \frac{x^{2}}{4}

    The result is: −x24+xsin⁡(x)+cos⁡(x)- \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}

  3. Add the constant of integration:

    −x24+xsin⁡(x)+cos⁡(x)+constant- \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}+ \mathrm{constant}

The answer is:

−x24+xsin⁡(x)+cos⁡(x)+constant- \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫x(cos⁡(x)−12) dx=C−x24+xsin⁡(x)+cos⁡(x)\int x \left(\cos{\left(x \right)} - \frac{1}{2}\right)\, dx = C - \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}
Simplify
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    Use the examples entering the upper and lower limits of integration.

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