Solve ∫ (1/6+1/2):(2-1/3)-(1/2-frac{1}{6})*frac{6}{5}

$$\int ( \frac { 1 } { 6 } + \frac { 1 } { 2 } ) : ( 2 - \frac { 1 } { 3 } ) - ( \frac { 1 } { 2 } - \frac { 1 } { 6 } ) \times \frac { 6 } { 5 }$$

Evaluate

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Solution Steps

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

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

Since \frac{1}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

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

Add 1 and 3 to get 4.

$$\int \frac{\frac{4}{6}}{2-\frac{1}{3}}-\left(\frac{1}{2}-\frac{1}{6}\right)\times \frac{6}{5}\mathrm{d}x$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

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

Convert 2 to fraction \frac{6}{3}.

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

Since \frac{6}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 1 from 6 to get 5.

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

Divide \frac{2}{3} by \frac{5}{3} by multiplying \frac{2}{3} by the reciprocal of \frac{5}{3}.

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

Multiply \frac{2}{3} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 3 in both numerator and denominator.

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

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

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

Since \frac{3}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\int \frac{2}{5}-\frac{3-1}{6}\times \frac{6}{5}\mathrm{d}x$$

Subtract 1 from 3 to get 2.

$$\int \frac{2}{5}-\frac{2}{6}\times \frac{6}{5}\mathrm{d}x$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\int \frac{2}{5}-\frac{1}{3}\times \frac{6}{5}\mathrm{d}x$$

Multiply \frac{1}{3} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\int \frac{2}{5}-\frac{1\times 6}{3\times 5}\mathrm{d}x$$

Do the multiplications in the fraction \frac{1\times 6}{3\times 5}.

$$\int \frac{2}{5}-\frac{6}{15}\mathrm{d}x$$

Reduce the fraction \frac{6}{15} to lowest terms by extracting and canceling out 3.

$$\int \frac{2}{5}-\frac{2}{5}\mathrm{d}x$$

Subtract \frac{2}{5} from \frac{2}{5} to get 0.

$$\int 0\mathrm{d}x$$

Find the integral of 0 using the table of common integrals rule \int a\mathrm{d}x=ax.

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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.

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Differentiate w.r.t. x

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