Solve ∫ (from 0 to 2) of (54.38x)(18/25x)

$$\int _ { 0 } ^ { 2 } ( 54.38 x ) ( 18 / 25 x )$$

Evaluate

$$104.4096$$

Solution Steps

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

$$\int _{0}^{2}54.38x^{2}\times \frac{18}{25}\mathrm{d}x$$

Convert decimal number 54.38 to fraction \frac{5438}{100}. Reduce the fraction \frac{5438}{100} to lowest terms by extracting and canceling out 2.

$$\int _{0}^{2}\frac{2719}{50}x^{2}\times \frac{18}{25}\mathrm{d}x$$

Multiply \frac{2719}{50} times \frac{18}{25} by multiplying numerator times numerator and denominator times denominator.

$$\int _{0}^{2}\frac{2719\times 18}{50\times 25}x^{2}\mathrm{d}x$$

Do the multiplications in the fraction \frac{2719\times 18}{50\times 25}.

$$\int _{0}^{2}\frac{48942}{1250}x^{2}\mathrm{d}x$$

Reduce the fraction \frac{48942}{1250} to lowest terms by extracting and canceling out 2.

$$\int _{0}^{2}\frac{24471}{625}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{24471x^{2}}{625}\mathrm{d}x$$

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

$$\frac{24471\int x^{2}\mathrm{d}x}{625}$$

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}.

$$\frac{8157x^{3}}{625}$$

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.

$$\frac{8157}{625}\times 2^{3}-\frac{8157}{625}\times 0^{3}$$

Simplify.

$$\frac{65256}{625}$$