in Calculus recategorized by
491 views
4 votes
4 votes
Let $\frac{d}{dx} [f(x)] = \frac{e^{sinx}}{x} , x > 0 .$

If $\int_{1}^{4}(\frac{2e^{sinx^{2}}}{x}) dx = f(k) - f(1)$ where limits of integration is from $1$ to $4$ , then $k =?$
in Calculus recategorized by
491 views

1 Answer

2 votes
2 votes

Let x2 = t,     Limit value  x = [ 1 , 4 ]  => t = [ 1 , 16 ] 

And  2x dx = dt

=>  2dx / x = dt / x2 = dt /  t.

   ∫ (2 esinx^2 / x) dx

= ∫  (esin t ) / t dt 

= ∫ F ' ( t) dt                      // d/dx [f(x)] = esinx / x , x > 0

= [F(t) ]116

= F(16 ) - F(1) 

Ans - k= 16.