Solve ∫ (from 0 to 3) of Lambda(6t)dt

$$\int _ { 0 } ^ { 3 } \Lambda ( 6 t ) d t$$

Evaluate

$$27\Lambda$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6\Lambda t\mathrm{d}t$$

Factor out the constant using \int af\left(t\right)\mathrm{d}t=a\int f\left(t\right)\mathrm{d}t.

$$6\Lambda \int t\mathrm{d}t$$

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

$$6\Lambda \times \frac{t^{2}}{2}$$

Simplify.

$$3\Lambda t^{2}$$

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.

$$3\Lambda \times 3^{2}-3\Lambda \times 0^{2}$$

Simplify.

$$27\Lambda$$

Differentiate w.r.t. Λ

$$27$$