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Solving integrals/ (2/cos²x-3sin²x)dx

Integral of (2/cos²x-3sin²x)dx dx

The solution

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  1                         
  /                         
 |                          
 |  /   2           2   \   
 |  |------- - 3*sin (x)| dx
 |  |   2               |   
 |  \cos (x)            /   
 |                          
/                           
0

$$\int\limits_{0}^{1} \left(- 3 \sin^{2}{\left(x \right)} + \frac{2}{\cos^{2}{\left(x \right)}}\right)\, dx$$

Integral(2/cos(x)^2 - 3*sin(x)^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 3 \sin^{2}{\left(x \right)}\right)\, dx = - 3 \int \sin^{2}{\left(x \right)}\, dx$
  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 $\frac{du}{2}$ :
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du$
        
        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: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
  So, the result is: $- \frac{3 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{\cos^{2}{\left(x \right)}}\, dx = 2 \int \frac{1}{\cos^{2}{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  So, the result is: $\frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}$
The result is: $- \frac{3 x}{2} + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \sin{\left(2 x \right)}}{4}$
Now simplify:

$- \frac{3 x}{2} + \frac{3 \sin{\left(2 x \right)}}{4} + 2 \tan{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  3   2*sin(1)   3*cos(1)*sin(1)
- - + -------- + ---------------
  2    cos(1)           2

$$- \frac{3}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{2 \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$

  3   2*sin(1)   3*cos(1)*sin(1)
- - + -------- + ---------------
  2    cos(1)           2

$$- \frac{3}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{2 \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$

-3/2 + 2*sin(1)/cos(1) + 3*cos(1)*sin(1)/2

Numerical answer [src]

2.29678851942907

2.29678851942907

The graph

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

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Integral of (2/cos²x-3sin²x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution