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Integral of -cos(2x-1) dx

The solution

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  1                 
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 |  -cos(2*x - 1) dx
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0

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

Integral(-cos(2*x - 1), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \cos{\left(2 x - 1 \right)}\right)\, dx = - \int \cos{\left(2 x - 1 \right)}\, dx$
1. Let $u = 2 x - 1$ .
  
  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:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
    1. 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:
  
  $\frac{\sin{\left(2 x - 1 \right)}}{2}$
So, the result is: $- \frac{\sin{\left(2 x - 1 \right)}}{2}$
Now simplify:

$- \frac{\sin{\left(2 x - 1 \right)}}{2}$
Add the constant of integration:

$- \frac{\sin{\left(2 x - 1 \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\sin{\left(2 x - 1 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-sin(1)

$$- \sin{\left(1 \right)}$$

-sin(1)

$$- \sin{\left(1 \right)}$$

-sin(1)

Numerical answer [src]

-0.841470984807897

-0.841470984807897

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

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Identical expressions

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Integral of -cos(2x-1) dx

Limits of integration:

The graph:

Piecewise:

The solution