Solve ∫ (from 0 to 2) of (5438x)(18/25x)

$$\int _ { 0 } ^ { 2 } ( 5438 x ) ( 18 / 25 x )$$

Evaluate

$$\frac{261024}{25}=10440.96$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{0}^{2}5438x^{2}\times \frac{18}{25}\mathrm{d}x$$

Express 5438\times \frac{18}{25} as a single fraction.

$$\int _{0}^{2}\frac{5438\times 18}{25}x^{2}\mathrm{d}x$$

Multiply 5438 and 18 to get 97884.

$$\int _{0}^{2}\frac{97884}{25}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{97884x^{2}}{25}\mathrm{d}x$$

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

$$\frac{97884\int x^{2}\mathrm{d}x}{25}$$

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

$$\frac{32628x^{3}}{25}$$

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.

$$\frac{32628}{25}\times 2^{3}-\frac{32628}{25}\times 0^{3}$$

Simplify.

$$\frac{261024}{25}$$