Solve ∫ (from 2 to 5) of ([2000*9.8)x] wrt x

$$\int _ { 2 } ^ { 5 } ( [ 2000 \times 9.8 ) x ] d x$$

Evaluate

$$205800$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{49\times 2000x}{5}\mathrm{d}x$$

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

$$\frac{49\times 2000}{5}\int x\mathrm{d}x$$

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

$$\frac{49\times 2000}{5}\times \frac{x^{2}}{2}$$

Simplify.

$$9800x^{2}$$

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.

$$9800\times 5^{2}-9800\times 2^{2}$$

Simplify.

$$205800$$