Solve ∫ (from a to b) of 8t^3dt

$$\int_{ a }^{ b } 8 { t }^{ 3 } d t$$

Evaluate

$$2\left(b^{4}-a^{4}\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 8t^{3}\mathrm{d}t$$

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

$$8\int t^{3}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{3}\mathrm{d}t with \frac{t^{4}}{4}. Multiply 8 times \frac{t^{4}}{4}.

$$2t^{4}$$

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.

$$2b^{4}-2a^{4}$$

Simplify.

$$-2a^{4}+2b^{4}$$

Differentiate w.r.t. a

$$-8a^{3}$$