Question

2ˣ + 1/√x²-9,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

we have to integrate $\int{\frac{2x+1}{\sqrt{x^2-9}}}\,dx$

$\int{\frac{2x+1}{\sqrt{x^2-9}}}\,dx$

= $\int{\frac{2x}{\sqrt{x^2-9}}}\,dx+\int{\frac{1}{\sqrt{x^2-9}}}\,dx$

if we assume f(x) = x² - 9 , then f'(x) = 2x
hence, $\int{\frac{2x}{\sqrt{x^2-9}}}\,dx=\int{\frac{f'(x)}{\sqrt{f(x)}}}\,dx$
we know, $\int{\frac{f'(x)}{\sqrt{f(x)}}}\,dx=2\sqrt{f(x)}$

and also we know,
$\int{\frac{1}{\sqrt{x^2-a^2}}}\,dx=log|\sqrt{x^2-a^2}+x|$


now, $2\sqrt{x^2-9}+log|\sqrt{x^2-9}+x|+C$