Question

$\int e^x\left[\log \right(x+\sqrt{x^2-1})+\frac{d}{dx}(\cos h^{-1}x\left)\right]dx=$

• A. $e^x\sin h^{-1}x+c$
• B. $e^x\log (x+\sqrt{x^2+1})+c$
• C. $e^x{\cosh }^{-1}x+c$
• D. $-e^x{\sinh }^{-1}x+c$

Answer 1

Answer: (C)

$e^x{\cosh }^{-1}x+c$

$\int e^x\left[\log \right(x+\sqrt{x^2-1})+\frac{d}{dx}({\cosh }^{-1}x\left)\right]dx$

$\int e^x[{\cosh }^{-1}x+\frac{d}{dx}({\cosh }^{-1}x\left)\right]dx$

we know that

$\log (x+\sqrt{x^2-1})={\cosh }^{-1}x$

so it is in the form of

$\int e^n\left[f\right(x)+f^1(x\left)\right]dx$

$=e^{x}f\left(x\right)+c$

so,

$\int e^n\left(\log \right(x+\sqrt{x^2+1})+\frac{d}{dx}(cos{h}^{-1}x\left)\right]dx$

$=e^x.{\cosh }^{-1}x+c$