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1/(2-x)
Solving integrals/ 1/(2-x)

Integral of 1/(2-x) dx

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The solution

You have entered [src] 
  1         
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∫0112−x dx\int\limits_{0}^{1} \frac{1}{2 - x}\, dx 
Integral(1/(2 - x), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2−xu = 2 - x.

      Then let du=−dxdu = - 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⁡(2−x)- \log{\left(2 - x \right)}

    Method #2

    1. Rewrite the integrand:

      12−x=−1x−2\frac{1}{2 - x} = - \frac{1}{x - 2}

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

      ∫(−1x−2) dx=−∫1x−2 dx\int \left(- \frac{1}{x - 2}\right)\, dx = - \int \frac{1}{x - 2}\, dx

      1. Let u=x−2u = x - 2.

        Then let du=dxdu = 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−2)\log{\left(x - 2 \right)}

      So, the result is: −log⁡(x−2)- \log{\left(x - 2 \right)}

    Method #3

    1. Rewrite the integrand:

      12−x=−1x−2\frac{1}{2 - x} = - \frac{1}{x - 2}

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

      ∫(−1x−2) dx=−∫1x−2 dx\int \left(- \frac{1}{x - 2}\right)\, dx = - \int \frac{1}{x - 2}\, dx

      1. Let u=x−2u = x - 2.

        Then let du=dxdu = 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−2)\log{\left(x - 2 \right)}

      So, the result is: −log⁡(x−2)- \log{\left(x - 2 \right)}

  2. Add the constant of integration:

    −log⁡(2−x)+constant- \log{\left(2 - x \right)}+ \mathrm{constant}

The answer is:

−log⁡(2−x)+constant- \log{\left(2 - x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                         
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 |   1                      
 | ----- dx = C - log(2 - x)
 | 2 - x                    
 |                          
/
∫12−x dx=C−log⁡(2−x)\int \frac{1}{2 - x}\, dx = C - \log{\left(2 - x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.5
Plot the graph f(x)
The answer [src] 
log(2)
log⁡(2)\log{\left(2 \right)} 
=
= 
log(2)
log⁡(2)\log{\left(2 \right)} 
log(2)
Numerical answer [src] 
0.693147180559945
0.693147180559945
The graph
Integral of 1/(2-x) dx

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

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