Solve ∫ _{1/5}^6-sqrt{11/10}(1.2-x) wrt x

$$\int _ { \frac { 1 } { 5 } } ^ { \frac { 6 - \sqrt { 11 } } { 10 } } ( 1.2 - x ) d x$$

Evaluate

$$-\frac{3\sqrt{11}}{50}+0.265\approx 0.066002513$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1.2-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1.2\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1.2\mathrm{d}x-\int x\mathrm{d}x$$

Find the integral of 1.2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{6x}{5}-\int x\mathrm{d}x$$

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 -1 times \frac{x^{2}}{2}.

$$\frac{6x}{5}-\frac{x^{2}}{2}$$

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.

$$1.2\times \frac{1}{10}\left(6-11^{\frac{1}{2}}\right)-\frac{1}{2}\times \left(\frac{1}{10}\left(6-11^{\frac{1}{2}}\right)\right)^{2}-\left(1.2\times \frac{1}{5}-\frac{\left(\frac{1}{5}\right)^{2}}{2}\right)$$

Simplify.

$$\frac{53}{200}-\frac{3\sqrt{11}}{50}$$