Mister Exam

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Similar expressions
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Solving integrals/ (sin3x-1/cos^2x)

Integral of (sin3x-1/cos^2x) dx

The solution

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

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

Simplify

Detail solution

Integrate term-by-term:
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin{\left(u \right)}}{9}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{3}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 x \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx = - \int \frac{1}{\cos^{2}{\left(x \right)}}\, dx$
  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: $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
The result is: $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\cos{\left(3 x \right)}}{3}$
Now simplify:

$- \frac{\cos{\left(3 x \right)}}{3} - \tan{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Simplify

Numerical answer [src]

0.0

0.0

The graph

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

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Similar expressions

Integral of (sin3x-1/cos^2x) dx

Limits of integration:

The graph:

Piecewise:

The solution