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Identical expressions
(sin(pi*x)-sqrt(x)+ one)
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( sinus of ( Pi multiply by x) minus square root of (x) plus one)
(sin(pi*x)-√(x)+1)
(sin(pix)-sqrt(x)+1)
sinpix-sqrtx+1
(sin(pi*x)-sqrt(x)+1)dx
Similar expressions
(sin(pi*x)-sqrt(x)-1)
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Solving integrals/ (sin(pi*x)-sqrt(x)+1)

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

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

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:
    
    $\int \left(- \sqrt{x}\right)\, dx = - \int \sqrt{x}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$
    So, the result is: $- \frac{2 x^{\frac{3}{2}}}{3}$
  1. Let $u = \pi x$ .
    
    Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
    
    $\int \frac{\sin{\left(u \right)}}{\pi}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}$
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      So, the result is: $- \frac{\cos{\left(u \right)}}{\pi}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(\pi x \right)}}{\pi}$
  The result is: $- \frac{2 x^{\frac{3}{2}}}{3} - \frac{\cos{\left(\pi x \right)}}{\pi}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}$
Add the constant of integration:

$- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}$

The answer is:

$- \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                                         3/2            
 | /              ___    \              2*x      cos(pi*x)
 | \sin(pi*x) - \/ x  + 1/ dx = C + x - ------ - ---------
 |                                        3          pi   
/

$$\int \left(\left(- \sqrt{x} + \sin{\left(\pi x \right)}\right) + 1\right)\, dx = C - \frac{2 x^{\frac{3}{2}}}{3} + x - \frac{\cos{\left(\pi x \right)}}{\pi}$$

Simplify

The graph

Plot the graph f(x)

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.

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

Limits of integration:

The graph:

Piecewise:

The solution