Integral von 11x2\frac{1}{1 - x^{2}}

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Ihr Beitrag

Finden Sie 11x2dx\int \frac{1}{1 - x^{2}}\, dx.

Lösung

Perform partial fraction decomposition (steps can be seen »):

11x2dx=(12(x+1)12(x1))dx{\color{red}{\int{\frac{1}{1 - x^{2}} d x}}} = {\color{red}{\int{\left(\frac{1}{2 \left(x + 1\right)} - \frac{1}{2 \left(x - 1\right)}\right)d x}}}

Integrate term by term:

(12(x+1)12(x1))dx=(12(x1)dx+12(x+1)dx){\color{red}{\int{\left(\frac{1}{2 \left(x + 1\right)} - \frac{1}{2 \left(x - 1\right)}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{2 \left(x - 1\right)} d x} + \int{\frac{1}{2 \left(x + 1\right)} d x}\right)}}

Apply the constant multiple rule cf(x)dx=cf(x)dx\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx with c=12c=\frac{1}{2} and f(x)=1x+1f{\left(x \right)} = \frac{1}{x + 1}:

12(x1)dx+12(x+1)dx=12(x1)dx+(1x+1dx2)- \int{\frac{1}{2 \left(x - 1\right)} d x} + {\color{red}{\int{\frac{1}{2 \left(x + 1\right)} d x}}} = - \int{\frac{1}{2 \left(x - 1\right)} d x} + {\color{red}{\left(\frac{\int{\frac{1}{x + 1} d x}}{2}\right)}}

Let u=x+1u=x + 1.

Then du=(x+1)dx=1dxdu=\left(x + 1\right)^{\prime }dx = 1 dx (steps can be seen »), and we have that dx=dudx = du.

Deshalb,

12(x1)dx+1x+1dx2=12(x1)dx+1udu2- \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{{\color{red}{\int{\frac{1}{x + 1} d x}}}}{2} = - \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2}

The integral of 1u\frac{1}{u} is 1udu=ln(u)\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}:

12(x1)dx+1udu2=12(x1)dx+ln(u)2- \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = - \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}

Recall that u=x+1u=x + 1:

ln(u)212(x1)dx=ln((x+1))212(x1)dx\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} - \int{\frac{1}{2 \left(x - 1\right)} d x} = \frac{\ln{\left(\left|{{\color{red}{\left(x + 1\right)}}}\right| \right)}}{2} - \int{\frac{1}{2 \left(x - 1\right)} d x}

Apply the constant multiple rule cf(x)dx=cf(x)dx\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx with c=12c=\frac{1}{2} and f(x)=1x1f{\left(x \right)} = \frac{1}{x - 1}:

ln(x+1)212(x1)dx=ln(x+1)2(1x1dx2)\frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - {\color{red}{\int{\frac{1}{2 \left(x - 1\right)} d x}}} = \frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - {\color{red}{\left(\frac{\int{\frac{1}{x - 1} d x}}{2}\right)}}

Let u=x1u=x - 1.

Then du=(x1)dx=1dxdu=\left(x - 1\right)^{\prime }dx = 1 dx (steps can be seen »), and we have that dx=dudx = du.

The integral can be rewritten as

ln(x+1)21x1dx2=ln(x+1)21udu2\frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{x - 1} d x}}}}{2} = \frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2}

The integral of 1u\frac{1}{u} is 1udu=ln(u)\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}:

ln(x+1)21udu2=ln(x+1)2ln(u)2\frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = \frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}

Recall that u=x1u=x - 1:

ln(x+1)2ln(u)2=ln(x+1)2ln((x1))2\frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} = \frac{\ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{\ln{\left(\left|{{\color{red}{\left(x - 1\right)}}}\right| \right)}}{2}

Deshalb,

11x2dx=ln(x1)2+ln(x+1)2\int{\frac{1}{1 - x^{2}} d x} = - \frac{\ln{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\ln{\left(\left|{x + 1}\right| \right)}}{2}

Vereinfachen:

11x2dx=ln(x1)+ln(x+1)2\int{\frac{1}{1 - x^{2}} d x} = \frac{- \ln{\left(\left|{x - 1}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)}}{2}

Fügen Sie die Integrationskonstante hinzu:

11x2dx=ln(x1)+ln(x+1)2+C\int{\frac{1}{1 - x^{2}} d x} = \frac{- \ln{\left(\left|{x - 1}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)}}{2}+C

Answer: 11x2dx=ln(x1)+ln(x+1)2+C\int{\frac{1}{1 - x^{2}} d x}=\frac{- \ln{\left(\left|{x - 1}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)}}{2}+C