Question

1/√x²-x+5,Integrate the given w.r.t. x.

Answer 1

we have to integrate 1/√(x² - x + 5)

Let's rearrange the term (x² - x + 5).
e.g., x² - x + 5 = x² -2.1/2.x + (1/2)² - (1/2)² + 5
= (x - 1/2)² - 1/4 + 5
= (x - 1/2)² + 19/4
= (x - 1/2)² + (√19/2)²

now,
$\int{\frac{1}{\sqrt{x^2-x+5}}}\,dx\\\\\\=\int{\frac{1}{\sqrt{(x-1/2)^2+(\sqrt{19}/2)^2}}}\,dx$

we know, $\int{\frac{1}{\sqrt{x^2+a^2}}}\,dx=ln|\sqrt{x^2+a^2}+x|$

so, $ln|\sqrt{(x-1/2)^2+(\sqrt{19}/2)^2}+(x+1/2)|+C$

=$ln|\sqrt{x^2-x+5}+(x+1/2)|+C$