Solve ∫ (from 0 to x) of t^2f(t)dt

$$\int _ { 0 } ^ { x } t ^ { 2 } f ( t ) d t$$

Evaluate

$$\frac{fx^{4}}{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int _{0}^{x}t^{3}f\mathrm{d}t$$

Evaluate the indefinite integral first.

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

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

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

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}.

$$f\times \frac{t^{4}}{4}$$

Simplify.

$$\frac{ft^{4}}{4}$$

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.

$$\frac{1}{4}fx^{4}-\frac{1}{4}f\times 0^{4}$$

Simplify.

$$\frac{fx^{4}}{4}$$

Differentiate w.r.t. x

$$fx^{3}$$