Virtual Gate Test Series: Calculus - Integration

Question

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 $k =?$

Answer 1

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

And  2x dx = dt

=>  2dx / x = dt / x² = dt /  t.

   ∫ (2 e^{sinx^2} / x) dx

= ∫  (e^{sin t} ) / t dt 

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

= [F(t) ]₁¹⁶

⁼ F(16 ) - F(1) 

Ans - k= 16.