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Identical expressions
√2tgx+1cos^2x
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√2tgx+1cos^2xdx
Similar expressions
√2tgx-1cos^2x

Integral of √2tgx+1cos^2x dx

The solution

You have entered [src]

  1                            
  /                            
 |                             
 |  /  __________      2   \   
 |  \\/ 2*tan(x)  + cos (x)/ dx
 |                             
/                              
0

$$\int\limits_{0}^{1} \left(\sqrt{2 \tan{\left(x \right)}} + \cos^{2}{\left(x \right)}\right)\, dx$$

Integral(sqrt(2*tan(x)) + cos(x)^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\sqrt{2} \int \sqrt{\tan{\left(x \right)}}\, dx$
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: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \sqrt{2} \int \sqrt{\tan{\left(x \right)}}\, dx$
Add the constant of integration:

$\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \sqrt{2} \int \sqrt{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \sqrt{2} \int \sqrt{\tan{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                         /             
 |                                                         |              
 | /  __________      2   \          x   sin(2*x)     ___  |   ________   
 | \\/ 2*tan(x)  + cos (x)/ dx = C + - + -------- + \/ 2 * | \/ tan(x)  dx
 |                                   2      4              |              
/                                                         /

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

Simplify

The answer [src]

  1                                
  /                                
 |                                 
 |  /   2        ___   ________\   
 |  \cos (x) + \/ 2 *\/ tan(x) / dx
 |                                 
/                                  
0

$$\int\limits_{0}^{1} \left(\cos^{2}{\left(x \right)} + \sqrt{2} \sqrt{\tan{\left(x \right)}}\right)\, dx$$

  1                                
  /                                
 |                                 
 |  /   2        ___   ________\   
 |  \cos (x) + \/ 2 *\/ tan(x) / dx
 |                                 
/                                  
0

$$\int\limits_{0}^{1} \left(\cos^{2}{\left(x \right)} + \sqrt{2} \sqrt{\tan{\left(x \right)}}\right)\, dx$$

Integral(cos(x)^2 + sqrt(2)*sqrt(tan(x)), (x, 0, 1))

Numerical answer [src]

1.75587940481822

1.75587940481822

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

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Integral of √2tgx+1cos^2x dx

Limits of integration:

The graph:

Piecewise:

The solution