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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                           
  /                           
 |                            
 |  /              ___    \   
 |  \sin(pi*x) - \/ x  + 1/ dx
 |                            
/                             
-1                            
$$\int\limits_{-1}^{1} \left(\left(- \sqrt{x} + \sin{\left(\pi x \right)}\right) + 1\right)\, dx$$
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:

        1. The integral of is when :

        So, the result is:

      1. Let .

        Then let and substitute :

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

          1. The integral of sine is negative cosine:

          So, the result is:

        Now substitute back in:

      The result is:

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

    The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                       
 |                                         3/2            
 | /              ___    \              2*x      cos(pi*x)
 | \sin(pi*x) - \/ x  + 1/ dx = C + x - ------ - ---------
 |                                        3          pi   
/                                                         
$$\int \left(\left(- \sqrt{x} + \sin{\left(\pi x \right)}\right) + 1\right)\, dx = C - \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}$$
The graph
The answer [src]
4   2*I
- - ---
3    3 
$$\frac{4}{3} - \frac{2 i}{3}$$
=
=
4   2*I
- - ---
3    3 
$$\frac{4}{3} - \frac{2 i}{3}$$
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.