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

$$\int _ { 1 } ^ { \frac { \pi } { 2 } } - ( x - 1 ) ( 2 x - 11 ) d x =$$

Evaluate

$$\frac{13\pi ^{2}}{8}-\frac{\pi ^{3}}{12}-\frac{11\pi }{2}+\frac{31}{6}\approx 1.342157834$$

Solution Steps

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

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

The opposite of -1 is 1.

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

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

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

Combine 11x and 2x to get 13x.

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

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.

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

Simplify.

$$-\frac{\pi ^{3}}{12}+\frac{13\pi ^{2}}{8}-\frac{11\pi }{2}+\frac{31}{6}$$