Integrals with \(e^{x}\) Evaluate the following integrals

\(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x\)

Solution 17E Step-1 Now , we have to evaluate ex dx.……………(1) x x x e dx., value is the area under the curve f(x) = e . So , here x x ex the integrand x is a rational function it is not defined at denominator equal to zero. Here the denominator is root value , under the root the values are positive , and here x =/ 0. Let , x t…………(2) Now , differentiate x = t both sides with respect to x , then we have d ( x) = d (t) dx dx d ( ) = dt dx dx d 1/2 dt 1/2 dx (x ) = dx , since = (x ). (1 ) x(1/2)1= dt , since d x = nx n1 2 dx dx ( 2 )x (1/2)= dx. 1 dt 2x = dx . 1 x dx = 2 dt…………….(3) Step-2 from(1), (2) and (3) , replace ‘ x’ wih ‘t’ , and ‘ x dx ‘ with ‘dt’. Now , the given integral becomes; ex dx = e(2dt) x = 2 edt, since Cx (d) = C x dx. t t t = 2 e +C , since e dt = e . x = 2e +C , since t = x. ex x Therefore , x dx = 2e +C.