Solve u ( t - a ) = ∫ (from 0 to t) of delta(t-a)

$$u ( t - a ) = \int _ { 0 } ^ { t } \delta ( t - a )$$

Evaluate

$$t\delta \left(t-a\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \delta \left(t-a\right)\mathrm{d}x$$

Find the integral of \delta \left(t-a\right) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\delta \left(t-a\right)x$$

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.

$$\delta \left(t-a\right)t-\delta \left(t-a\right)\times 0$$

Simplify.

$$\delta \left(t-a\right)t$$

Differentiate w.r.t. δ

$$t\left(t-a\right)$$