Solve ∫ (1/2-1/3)(2-1/5)div(1-1/3)=

$$\int ( \frac { 1 } { 2 } - \frac { 1 } { 3 } ) ( 2 - \frac { 1 } { 5 } ) \div ( 1 - \frac { 1 } { 3 } ) =$$

Evaluate

$$\frac{9x}{20}+С$$

Solution Steps

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\int \frac{\left(\frac{3}{6}-\frac{2}{6}\right)\left(2-\frac{1}{5}\right)}{1-\frac{1}{3}}\mathrm{d}x$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$\int \frac{\frac{3-2}{6}\left(2-\frac{1}{5}\right)}{1-\frac{1}{3}}\mathrm{d}x$$

Subtract 2 from 3 to get 1.

$$\int \frac{\frac{1}{6}\left(2-\frac{1}{5}\right)}{1-\frac{1}{3}}\mathrm{d}x$$

Convert 2 to fraction \frac{10}{5}.

$$\int \frac{\frac{1}{6}\left(\frac{10}{5}-\frac{1}{5}\right)}{1-\frac{1}{3}}\mathrm{d}x$$

Since \frac{10}{5} and \frac{1}{5} have the same denominator, subtract them by subtracting their numerators.

$$\int \frac{\frac{1}{6}\times \frac{10-1}{5}}{1-\frac{1}{3}}\mathrm{d}x$$

Subtract 1 from 10 to get 9.

$$\int \frac{\frac{1}{6}\times \frac{9}{5}}{1-\frac{1}{3}}\mathrm{d}x$$

Multiply \frac{1}{6} times \frac{9}{5} by multiplying numerator times numerator and denominator times denominator.

$$\int \frac{\frac{1\times 9}{6\times 5}}{1-\frac{1}{3}}\mathrm{d}x$$

Do the multiplications in the fraction \frac{1\times 9}{6\times 5}.

$$\int \frac{\frac{9}{30}}{1-\frac{1}{3}}\mathrm{d}x$$

Reduce the fraction \frac{9}{30} to lowest terms by extracting and canceling out 3.

$$\int \frac{\frac{3}{10}}{1-\frac{1}{3}}\mathrm{d}x$$

Convert 1 to fraction \frac{3}{3}.

$$\int \frac{\frac{3}{10}}{\frac{3}{3}-\frac{1}{3}}\mathrm{d}x$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\int \frac{\frac{3}{10}}{\frac{3-1}{3}}\mathrm{d}x$$

Subtract 1 from 3 to get 2.

$$\int \frac{\frac{3}{10}}{\frac{2}{3}}\mathrm{d}x$$

Divide \frac{3}{10} by \frac{2}{3} by multiplying \frac{3}{10} by the reciprocal of \frac{2}{3}.

$$\int \frac{3}{10}\times \frac{3}{2}\mathrm{d}x$$

Multiply \frac{3}{10} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\int \frac{3\times 3}{10\times 2}\mathrm{d}x$$

Do the multiplications in the fraction \frac{3\times 3}{10\times 2}.

$$\int \frac{9}{20}\mathrm{d}x$$

Find the integral of \frac{9}{20} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{9x}{20}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{9x}{20}+С$$

Differentiate w.r.t. x

$$\frac{9}{20} = 0.45$$