What is the integral from-1 to 1 of x/(5-4x) ?

The integral from-1 to 1 of x/(5-4x) is -1/2+5/8 ln(3)

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integral from-1 to 1 of x/(5-4x)

\int_{- 1}^{1} \frac{x}{5 - 4 x} d x

- \frac{1}{2} + \frac{5}{8} ln (3)

Decimal

0.18663 \dots

Solution steps

\int_{- 1}^{1} \frac{x}{5 - 4 x} d x

Long division

\frac{x}{5 - 4 x} : - \frac{1}{4} + \frac{5}{4 ( - 4 x + 5 )}

= \int_{- 1}^{1} - \frac{1}{4} + \frac{5}{4 ( - 4 x + 5 )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{- 1}^{1} \frac{1}{4} d x + \int_{- 1}^{1} \frac{5}{4 ( - 4 x + 5 )} d x

\int_{- 1}^{1} \frac{1}{4} d x = \frac{1}{2}

\int_{- 1}^{1} \frac{5}{4 ( - 4 x + 5 )} d x = \frac{5}{8} ln (3)

= - \frac{1}{2} + \frac{5}{8} ln (3)

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What is the integral from-1 to 1 of x/(5-4x) ?
The integral from-1 to 1 of x/(5-4x) is -1/2+5/8 ln(3)