Solve ∫ (from 0 to 4) of (2.76x-0.3x^3) wrt x

$$\int_{ 0 }^{ 4 } (2.76x-0.3 { x }^{ 3 } ) d x$$

Evaluate

$$2.88$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{69x}{25}-\frac{3x^{3}}{10}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{69x}{25}\mathrm{d}x+\int -\frac{3x^{3}}{10}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{69\int x\mathrm{d}x}{25}-\frac{3\int x^{3}\mathrm{d}x}{10}$$

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 2.76 times \frac{x^{2}}{2}.

$$\frac{69x^{2}}{50}-\frac{3\int x^{3}\mathrm{d}x}{10}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -0.3 times \frac{x^{4}}{4}.

$$\frac{69x^{2}}{50}-\frac{3x^{4}}{40}$$

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.

$$\frac{69}{50}\times 4^{2}-\frac{3}{40}\times 4^{4}-\left(\frac{69}{50}\times 0^{2}-\frac{3}{40}\times 0^{4}\right)$$

Simplify.

$$\frac{72}{25}$$