Solve ∫ (from 0 to 4) of (-47+23.5x+33.437x)(0.5x)/2 wrt x

$$\int_{ 0 }^{ 4 } \frac{ (-47+23.5x+33.437x)(0.5x) }{ 2 } d x$$

Evaluate

$$209.664$$

Solution Steps

Combine 23.5x and 33.437x to get 56.937x.

$$\int _{0}^{4}\frac{\left(-47+56.937x\right)\times 0.5x}{2}\mathrm{d}x$$

Divide \left(-47+56.937x\right)\times 0.5x by 2 to get \left(-47+56.937x\right)\times 0.25x.

$$\int _{0}^{4}\left(-47+56.937x\right)\times 0.25x\mathrm{d}x$$

Use the distributive property to multiply -47+56.937x by 0.25.

$$\int _{0}^{4}\left(-11.75+14.23425x\right)x\mathrm{d}x$$

Use the distributive property to multiply -11.75+14.23425x by x.

$$\int _{0}^{4}-11.75x+14.23425x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{47x}{4}+\frac{56937x^{2}}{4000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{47x}{4}\mathrm{d}x+\int \frac{56937x^{2}}{4000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{47\int x\mathrm{d}x}{4}+\frac{56937\int x^{2}\mathrm{d}x}{4000}$$

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}. Multiply -11.75 times \frac{x^{2}}{2}.

$$-\frac{47x^{2}}{8}+\frac{56937\int x^{2}\mathrm{d}x}{4000}$$

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}. Multiply 14.23425 times \frac{x^{3}}{3}.

$$-\frac{47x^{2}}{8}+\frac{18979x^{3}}{4000}$$

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.

$$-\frac{47}{8}\times 4^{2}+\frac{18979}{4000}\times 4^{3}-\left(-\frac{47}{8}\times 0^{2}+\frac{18979}{4000}\times 0^{3}\right)$$

Simplify.

$$\frac{26208}{125}$$