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Solving integrals/ (x-1/2)*e^x

Integral of (x-1/2)*e^x dx

Limits of integration:

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The solution

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∫01ex(x−12) dx\int\limits_{0}^{1} e^{x} \left(x - \frac{1}{2}\right)\, dx 
Integral((x - 1/2)*E^x, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    ex(x−12)=xex−ex2e^{x} \left(x - \frac{1}{2}\right) = x e^{x} - \frac{e^{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)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

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

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

      1. The integral of the exponential function is itself.

        ∫ex dx=ex\int e^{x}\, dx = e^{x}

      Now evaluate the sub-integral.

    2. The integral of the exponential function is itself.

      ∫ex dx=ex\int e^{x}\, dx = e^{x}

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

      ∫(−ex2) dx=−∫ex dx2\int \left(- \frac{e^{x}}{2}\right)\, dx = - \frac{\int e^{x}\, dx}{2}

      1. The integral of the exponential function is itself.

        ∫ex dx=ex\int e^{x}\, dx = e^{x}

      So, the result is: −ex2- \frac{e^{x}}{2}

    The result is: xex−3ex2x e^{x} - \frac{3 e^{x}}{2}

  3. Now simplify:

    (x−32)ex\left(x - \frac{3}{2}\right) e^{x}

  4. Add the constant of integration:

    (x−32)ex+constant\left(x - \frac{3}{2}\right) e^{x}+ \mathrm{constant}

The answer is:

(x−32)ex+constant\left(x - \frac{3}{2}\right) e^{x}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
 |                          x       
 |            x          3*e       x
 | (x - 1/2)*E  dx = C - ---- + x*e 
 |                        2         
/
∫ex(x−12) dx=C+xex−3ex2\int e^{x} \left(x - \frac{1}{2}\right)\, dx = C + x e^{x} - \frac{3 e^{x}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
3   E
- - -
2   2
32−e2\frac{3}{2} - \frac{e}{2} 
=
= 
3   E
- - -
2   2
32−e2\frac{3}{2} - \frac{e}{2} 
3/2 - E/2
Numerical answer [src] 
0.140859085770477
0.140859085770477

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

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