# In the given equation as follows , use a table of integrals to find the indefini

In the given equation as follows , use a table of integrals to find the indefinite integral:
$\int xar\mathcal{s}c\left({x}^{2}+1\right)dx$
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Neelam Wainwright
In the given equation as follows , use a table of integrals to find the indefinite integral:-
$\int xar\mathcal{s}c\left({x}^{2}+1\right)dx$
Solution:
$\int x{\mathrm{sec}}^{-1}\left({x}^{2}+1\right)dx$
Let us substitute,
${x}^{2}+1=z$
$2xdx=dz$
Solution:
Now,
$\int \frac{1}{2}{\mathrm{sec}}^{-1}zdz$
$=\frac{1}{2}\left[{\mathrm{sec}}^{-1}z\int dz-\int \left[\frac{d}{dz}\left({\mathrm{sec}}^{-1}z\right)\int dz\right]\right]$
$=\frac{1}{2}\left[z{\mathrm{sec}}^{-1}z-\frac{1}{\sqrt{{z}^{2}-1}}z\right]$
$=\frac{1}{2}\left[\left({x}^{2}+1\right){\mathrm{sec}}^{-1}\left({x}^{2}+1\right)-\frac{{x}^{2}+1}{\sqrt{{\left({x}^{2}+1\right)}^{2}-1}}\right]+C$