Solve ∫ (from 1 to frac{ 11) of {2}}-left(x-1right)left(2x-11right) wrt x

$$\int_{ 1 }^{ \frac{ 11 }{ 2 } } - \left( x-1 \right) \left( 2x-11 \right) d x$$

Evaluate

$$\frac{243}{8}=30.375$$

Solution Steps

To find the opposite of x-1, find the opposite of each term.

$$\int _{1}^{\frac{11}{2}}\left(-x-\left(-1\right)\right)\left(2x-11\right)\mathrm{d}x$$

The opposite of -1 is 1.

$$\int _{1}^{\frac{11}{2}}\left(-x+1\right)\left(2x-11\right)\mathrm{d}x$$

Apply the distributive property by multiplying each term of -x+1 by each term of 2x-11.

$$\int _{1}^{\frac{11}{2}}-2x^{2}+11x+2x-11\mathrm{d}x$$

Combine 11x and 2x to get 13x.

$$\int _{1}^{\frac{11}{2}}-2x^{2}+13x-11\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -2x^{2}+13x-11\mathrm{d}x$$

Integrate the sum term by term.

$$\int -2x^{2}\mathrm{d}x+\int 13x\mathrm{d}x+\int -11\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-2\int x^{2}\mathrm{d}x+13\int x\mathrm{d}x+\int -11\mathrm{d}x$$

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 -2 times \frac{x^{3}}{3}.

$$-\frac{2x^{3}}{3}+13\int x\mathrm{d}x+\int -11\mathrm{d}x$$

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

$$-\frac{2x^{3}}{3}+\frac{13x^{2}}{2}+\int -11\mathrm{d}x$$

Find the integral of -11 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{2x^{3}}{3}+\frac{13x^{2}}{2}-11x$$

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.

$$-\frac{2}{3}\times \left(\frac{11}{2}\right)^{3}+\frac{13}{2}\times \left(\frac{11}{2}\right)^{2}-11\times \frac{11}{2}-\left(-\frac{2}{3}\times 1^{3}+\frac{13}{2}\times 1^{2}-11\right)$$

Simplify.

$$\frac{243}{8}$$