Mister Exam

Other calculators

(1-sin(x))/(x+cos(x))
Solving integrals/ (1-sin(x))/(x+cos(x))

Integral of (1-sin(x))/(x+cos(x)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1              
  /              
 |               
 |  1 - sin(x)   
 |  ---------- dx
 |  x + cos(x)   
 |               
/                
0
∫011−sin⁡(x)x+cos⁡(x) dx\int\limits_{0}^{1} \frac{1 - \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\, dx 
Integral((1 - sin(x))/(x + cos(x)), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+cos⁡(x)u = x + \cos{\left(x \right)}.

      Then let du=(1−sin⁡(x))dxdu = \left(1 - \sin{\left(x \right)}\right) dx and substitute dudu:

      ∫1u du\int \frac{1}{u}\, du

      1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

      Now substitute uu back in:

      log⁡(x+cos⁡(x))\log{\left(x + \cos{\left(x \right)} \right)}

    Method #2

    1. Rewrite the integrand:

      1−sin⁡(x)x+cos⁡(x)=−sin⁡(x)−1x+cos⁡(x)\frac{1 - \sin{\left(x \right)}}{x + \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)} - 1}{x + \cos{\left(x \right)}}

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

      ∫(−sin⁡(x)−1x+cos⁡(x)) dx=−∫sin⁡(x)−1x+cos⁡(x) dx\int \left(- \frac{\sin{\left(x \right)} - 1}{x + \cos{\left(x \right)}}\right)\, dx = - \int \frac{\sin{\left(x \right)} - 1}{x + \cos{\left(x \right)}}\, dx

      1. Let u=x+cos⁡(x)u = x + \cos{\left(x \right)}.

        Then let du=(1−sin⁡(x))dxdu = \left(1 - \sin{\left(x \right)}\right) dx and substitute −du- du:

        ∫(−1u) du\int \left(- \frac{1}{u}\right)\, du

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

          ∫1u du=−∫1u du\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: −log⁡(u)- \log{\left(u \right)}

        Now substitute uu back in:

        −log⁡(x+cos⁡(x))- \log{\left(x + \cos{\left(x \right)} \right)}

      So, the result is: log⁡(x+cos⁡(x))\log{\left(x + \cos{\left(x \right)} \right)}

  2. Add the constant of integration:

    log⁡(x+cos⁡(x))+constant\log{\left(x + \cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

log⁡(x+cos⁡(x))+constant\log{\left(x + \cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
 |                                    
 | 1 - sin(x)                         
 | ---------- dx = C + log(x + cos(x))
 | x + cos(x)                         
 |                                    
/
∫1−sin⁡(x)x+cos⁡(x) dx=C+log⁡(x+cos⁡(x))\int \frac{1 - \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\, dx = C + \log{\left(x + \cos{\left(x \right)} \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
log(1 + cos(1))
log⁡(cos⁡(1)+1)\log{\left(\cos{\left(1 \right)} + 1 \right)} 
=
= 
log(1 + cos(1))
log⁡(cos⁡(1)+1)\log{\left(\cos{\left(1 \right)} + 1 \right)} 
log(1 + cos(1))
Numerical answer [src] 
0.4319786996725
0.4319786996725
The graph
Integral of (1-sin(x))/(x+cos(x)) dx

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

    © Mister Exam – Calculator