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Solving integrals/ (sin(pi*x)-sqrt(x)+1)

Integral of (sin(pi*x)-sqrt(x)+1) dx

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The solution

You have entered [src] 
  1                           
  /                           
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 |  /              ___    \   
 |  \sin(pi*x) - \/ x  + 1/ dx
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∫−11((−x+sin⁡(πx))+1) dx\int\limits_{-1}^{1} \left(\left(- \sqrt{x} + \sin{\left(\pi x \right)}\right) + 1\right)\, dx 
Integral(sin(pi*x) - sqrt(x) + 1, (x, -1, 1))
Detail solution

  1. Integrate term-by-term:

    1. Integrate term-by-term:

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

        ∫(−x) dx=−∫x dx\int \left(- \sqrt{x}\right)\, dx = - \int \sqrt{x}\, dx

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

          ∫x dx=2x323\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}

        So, the result is: −2x323- \frac{2 x^{\frac{3}{2}}}{3}

      1. Let u=πxu = \pi x.

        Then let du=πdxdu = \pi dx and substitute duπ\frac{du}{\pi}:

        ∫sin⁡(u)π du\int \frac{\sin{\left(u \right)}}{\pi}\, du

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

          ∫sin⁡(u) du=∫sin⁡(u) duπ\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}

          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: −cos⁡(u)π- \frac{\cos{\left(u \right)}}{\pi}

        Now substitute uu back in:

        −cos⁡(πx)π- \frac{\cos{\left(\pi x \right)}}{\pi}

      The result is: −2x323−cos⁡(πx)π- \frac{2 x^{\frac{3}{2}}}{3} - \frac{\cos{\left(\pi x \right)}}{\pi}

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    The result is: −2x323+x−cos⁡(πx)π- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}

  2. Add the constant of integration:

    −2x323+x−cos⁡(πx)π+constant- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}

The answer is:

−2x323+x−cos⁡(πx)π+constant- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                       
 |                                         3/2            
 | /              ___    \              2*x      cos(pi*x)
 | \sin(pi*x) - \/ x  + 1/ dx = C + x - ------ - ---------
 |                                        3          pi   
/
∫((−x+sin⁡(πx))+1) dx=C−2x323+x−cos⁡(πx)π\int \left(\left(- \sqrt{x} + \sin{\left(\pi x \right)}\right) + 1\right)\, dx = C - \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}
Simplify
The graph
1.000.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
4   2*I
- - ---
3    3
43−2i3\frac{4}{3} - \frac{2 i}{3} 
=
= 
4   2*I
- - ---
3    3
43−2i3\frac{4}{3} - \frac{2 i}{3} 
4/3 - 2*i/3
Numerical answer [src] 
(1.33413269908838 - 0.665867300911624j)
(1.33413269908838 - 0.665867300911624j)

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

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