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Solving integrals/ dx/1+sin^2x

Integral of dx/1+sin^2x dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  /         2   \   
 |  \1.0 + sin (x)/ dx
 |                    
/                     
0

$$\int\limits_{0}^{1} \left(\sin^{2}{\left(x \right)} + 1.0\right)\, dx$$

Integral(1.0 + sin(x)^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Rewrite the integrand:
  
  $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    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: $- \frac{\sin{\left(2 x \right)}}{4}$
  The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1.0\, dx = 1.0 x$
The result is: $1.5 x - \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

$1.5 x - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$1.5 x - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 | /         2   \          sin(2*x)        
 | \1.0 + sin (x)/ dx = C - -------- + 1.5*x
 |                             4            
/

$$\int \left(\sin^{2}{\left(x \right)} + 1.0\right)\, dx = C + 1.5 x - \frac{\sin{\left(2 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      cos(1)*sin(1)
1.5 - -------------
            2

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + 1.5$$

      cos(1)*sin(1)
1.5 - -------------
            2

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + 1.5$$

1.5 - cos(1)*sin(1)/2

Numerical answer [src]

1.27267564329358

1.27267564329358

The graph

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

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Identical expressions

Similar expressions

Integral of dx/1+sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution