Solve ∫ (from 0 to frac { l) of {a}}(1-x) wrt x

$$\int _ { 0 } ^ { \frac { l } { a } } ( 1 - x ) d x$$

Evaluate

$$\lim_{x\to \frac{l}{a}}-\frac{x^{2}}{2}+x$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}x-\int x\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -1 times \frac{x^{2}}{2}.

$$x-\frac{x^{2}}{2}$$

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.

$$la^{-1}-\frac{1}{2}\left(la^{-1}\right)^{2}-\left(0-\frac{0^{2}}{2}\right)$$

Simplify.

$$\left\{\begin{matrix}\frac{l\left(-l+2a\right)}{2a^{2}},&\\\text{Indeterminate},&a=0\end{matrix}\right.$$

Differentiate w.r.t. l

$$\frac{\mathrm{d}}{\mathrm{d}l}(\lim_{x\to \frac{l}{a}}-\frac{x^{2}}{2}+x)$$