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Identical expressions
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Similar expressions
Cosxdx/2sinx-1

Integral of Cosxdx/2sinx+1 dx

The solution

You have entered [src]

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

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

Integral((cos(x)/2)*sin(x) + 1, (x, 0, pi/2))

Detail solution

Integrate term-by-term:
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
    
    $\int \left(- \frac{u}{2}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{4}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(x \right)}}{4}$
  Method #2
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{u}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{2}{\left(x \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x - \frac{\cos^{2}{\left(x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                        
 |                                     2   
 | /cos(x)           \              cos (x)
 | |------*sin(x) + 1| dx = C + x - -------
 | \  2              /                 4   
 |                                         
/

$$\int \left(\frac{\cos{\left(x \right)}}{2} \sin{\left(x \right)} + 1\right)\, dx = C + x - \frac{\cos^{2}{\left(x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   pi
- + --
4   2

$$\frac{1}{4} + \frac{\pi}{2}$$

1   pi
- + --
4   2

$$\frac{1}{4} + \frac{\pi}{2}$$

1/4 + pi/2

Numerical answer [src]

1.8207963267949

1.8207963267949

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

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Identical expressions

Similar expressions

Integral of Cosxdx/2sinx+1 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2