Solve ∫ (from 1 * 5 to 3 * 5) of (25t+9 t^2)dt

$$\int_{ 1 \cdot 5 }^{ 3 \cdot 5 } (25t+9 { t }^{ 2 } ) d t$$

Evaluate

$$12250$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 25t+9t^{2}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 25t\mathrm{d}t+\int 9t^{2}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$25\int t\mathrm{d}t+9\int t^{2}\mathrm{d}t$$

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}. Multiply 25 times \frac{t^{2}}{2}.

$$\frac{25t^{2}}{2}+9\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply 9 times \frac{t^{3}}{3}.

$$\frac{25t^{2}}{2}+3t^{3}$$

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.

$$\frac{25}{2}\times \left(3\times 5\right)^{2}+3\times \left(3\times 5\right)^{3}-\left(\frac{25}{2}\times 5^{2}+3\times 5^{3}\right)$$

Simplify.

$$12250$$