Solve ∫ (from 0 to 8) of (-133x)(-1/12x) wrt x

$$\int_{ 0 }^{ 8 } (-133x)(- \frac{ 1 }{ 12 } x) d x$$

Evaluate

$$\frac{17024}{9}\approx 1891.555555556$$

Solution Steps

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

$$\int _{0}^{8}-133x^{2}\left(-\frac{1}{12}\right)\mathrm{d}x$$

Express -133\left(-\frac{1}{12}\right) as a single fraction.

$$\int _{0}^{8}\frac{-133\left(-1\right)}{12}x^{2}\mathrm{d}x$$

Multiply -133 and -1 to get 133.

$$\int _{0}^{8}\frac{133}{12}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{133x^{2}}{12}\mathrm{d}x$$

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

$$\frac{133\int x^{2}\mathrm{d}x}{12}$$

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}.

$$\frac{133x^{3}}{36}$$

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.

$$\frac{133}{36}\times 8^{3}-\frac{133}{36}\times 0^{3}$$

Simplify.

$$\frac{17024}{9}$$