Solve ∫ (from n = 1 to 25) of 60+60n

$$\int _ { n = 1 } ^ { 25 } 60 + 60 n$$

Evaluate

$$20160$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 60+60n\mathrm{d}n$$

Integrate the sum term by term.

$$\int 60\mathrm{d}n+\int 60n\mathrm{d}n$$

Factor out the constant in each of the terms.

$$\int 60\mathrm{d}n+60\int n\mathrm{d}n$$

Find the integral of 60 using the table of common integrals rule \int a\mathrm{d}n=an.

$$60n+60\int n\mathrm{d}n$$

Since \int n^{k}\mathrm{d}n=\frac{n^{k+1}}{k+1} for k\neq -1, replace \int n\mathrm{d}n with \frac{n^{2}}{2}. Multiply 60 times \frac{n^{2}}{2}.

$$60n+30n^{2}$$

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.

$$60\times 25+30\times 25^{2}-\left(60\times 1+30\times 1^{2}\right)$$

Simplify.

$$20160$$