Solve ∫ (from 0 to 2) of (2.7x)(-x) wrt x

$$\int_{ 0 }^{ 2 } (2.7x)(-x) d x$$

Evaluate

$$-7.2$$

Solution Steps

Multiply 2.7 and -1 to get -2.7.

$$\int _{0}^{2}-2.7xx\mathrm{d}x$$

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

$$\int _{0}^{2}-2.7x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{27x^{2}}{10}\mathrm{d}x$$

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

$$-\frac{27\int x^{2}\mathrm{d}x}{10}$$

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}.

$$-\frac{9x^{3}}{10}$$

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.

$$-\frac{9}{10}\times 2^{3}+\frac{9}{10}\times 0^{3}$$

Simplify.

$$-\frac{36}{5}$$