# Hot Integral-5

**Calculus**Level 5

Let

\(\displaystyle f(x)=\int\limits_0^1 \frac{\sqrt{1-xt^2}}{\sqrt{1-t^2}}dt\)

Then, the following integral can be expressed as :

\(\displaystyle \int \frac{f(2015x^2)}{x^3}dx = -\frac{a}{bz^c}[\pi \left\{dz^f \;_4F_3(g,h,\frac{i}{k},\frac{j}{k};l,m,m;nz^o) -pz^q \log(r) + sz^t\log(-uz^v) + w\right\}]\) + constant

Evaluate \(a+b+c+d+f+g+h+i+j+k+l+m+n+o+p+q+r+s+t+u+v+w = ?\)

\(\textbf{Details and Assumptions}\)

1)\(a,b,c,d,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w\) are all positive integers.

2) \(_4F_3(.,.,.,.;.,.,.;.)\) is a \(\textbf{Hypergeometric function}\).

3) Everything is in the \(\textbf{simplest form}\) and \(r\) is square free.

\(\blacksquare\) This is a part of Hot Integrals

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