Solve ∫ (from e to e ^ 2) of +frac{1{f}-1}(u) wrt x

$$\int _ { e } ^ { e ^ { 2 } + \frac { 1 } { f } - 1 } ( u ) d x$$

Evaluate

$$\left(\lim_{x\to e^{2}-1+\frac{1}{f}}ux\right)-eu$$

Solution Steps

Evaluate the indefinite integral first.

$$\int u\mathrm{d}x$$

Find the integral of u using the table of common integrals rule \int a\mathrm{d}x=ax.

$$ux$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$u\left(e^{2}+f^{-1}-1\right)-ue$$

Simplify.

$$\left\{\begin{matrix}ue^{2}+\frac{u}{f}-u-ue,&\\\text{Indeterminate},&f=0\end{matrix}\right.$$

Differentiate w.r.t. u

$$\frac{\mathrm{d}}{\mathrm{d}u}(\lim_{x\to e^{2}-1+\frac{1}{f}}ux)-e$$