Question

Use fundamental theorem of calculus to find derivative of the function LOOK AT PHOTO

Answer 1

Let c > 0. Then split the integral at t = c to write

`f(x) = \displaystyle \int_(\ln(x))^(\frac1x) (t + \sin(t)) \, dt = \int_c^(\frac1x) (t + \sin(t)) \, dt - \int_c^(\ln(x)) (t + \sin(t)) \, dt`

By the FTC, the derivative is

`\displaystyle (df)/(dx) = \left(\frac1x + \sin\left(\frac1x\right)\right) (d)/(dx)\left[\frac1x\right] - (\ln(x) + \sin(\ln(x))) (d)/(dx)\left[\ln(x)\right] \\\\ = -\frac1{x^2} \left(\frac1x + \sin\left(\frac1x\right)\right) - \frac1x (\ln(x) + \sin(\ln(x))) \\\\ = -\frac1{x^3} - (\sin\left(\frac1x\right))/(x^2) - \frac{\ln(x)}x - \frac{\sin(\ln(x))}x \\\\ = -(1 + x\sin\left(\frac1x\right) + x^2\ln(x) + x^2 \sin(\ln(x)))/(x^3)`