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Identical expressions
√(one +sinx)²+(cos²x)dx
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Similar expressions
√(1-sinx)²+(cos²x)dx
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Integral of √(1+sinx)²+(cos²x)dx dx

The solution

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 -pi                               
 ----                              
  2                                
   /                               
  |                                
  |  /              2          \   
  |  |  ____________       2   |   
  |  \\/ 1 + sin(x)   + cos (x)/ dx
  |                                
 /                                 
 pi                                
 --                                
 2

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

Integral((sqrt(1 + sin(x)))^2 + cos(x)^2, (x, pi/2, -pi/2))

Detail solution

Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $x - \cos{\left(x \right)}$
1. Rewrite the integrand:
  
  $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
The result is: $\frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} - \cos{\left(x \right)}$
Add the constant of integration:

$\frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} - \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} - \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                                             
 | /              2          \                                 
 | |  ____________       2   |                   sin(2*x)   3*x
 | \\/ 1 + sin(x)   + cos (x)/ dx = C - cos(x) + -------- + ---
 |                                                  4        2 
/

$$\int \left(\left(\sqrt{\sin{\left(x \right)} + 1}\right)^{2} + \cos^{2}{\left(x \right)}\right)\, dx = C + \frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} - \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-3*pi
-----
  2

$$- \frac{3 \pi}{2}$$

-3*pi
-----
  2

$$- \frac{3 \pi}{2}$$

-3*pi/2

Numerical answer [src]

-4.71238898038469

-4.71238898038469

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

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Identical expressions

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Integral of √(1+sinx)²+(cos²x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution