Solve ∫ (from 0 to 2) of (120a-x)da

$$\int _ { 0 } ^ { 2 } ( 120 a - x ) d a$$

Evaluate

$$240-2x$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 120a-x\mathrm{d}a$$

Integrate the sum term by term.

$$\int 120a\mathrm{d}a+\int -x\mathrm{d}a$$

Factor out the constant in each of the terms.

$$120\int a\mathrm{d}a-\int x\mathrm{d}a$$

Since \int a^{k}\mathrm{d}a=\frac{a^{k+1}}{k+1} for k\neq -1, replace \int a\mathrm{d}a with \frac{a^{2}}{2}. Multiply 120 times \frac{a^{2}}{2}.

$$60a^{2}-\int x\mathrm{d}a$$

Find the integral of x using the table of common integrals rule \int a\mathrm{d}a=aa.

$$60a^{2}-xa$$

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.

$$60\times 2^{2}-x\times 2-\left(60\times 0^{2}-x\times 0\right)$$

Simplify.

$$240-2x$$

Differentiate w.r.t. x

$$-2$$