What is the integral from 2 to 4 of (2x+5)/(5-x) ?

The integral from 2 to 4 of (2x+5)/(5-x) is -4+15ln(3)

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integral from 2 to 4 of (2x+5)/(5-x)

\int_{2}^{4} \frac{2 x + 5}{5 - x} d x

- 4 + 15 ln (3)

Decimal

12.47918 \dots

Solution steps

\int_{2}^{4} \frac{2 x + 5}{5 - x} d x

Expand

\frac{2 x + 5}{5 - x} : \frac{2 x}{5 - x} + \frac{5}{5 - 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_{2}^{4} \frac{2 x}{5 - x} d x + \int_{2}^{4} \frac{5}{5 - x} d x

\int_{2}^{4} \frac{2 x}{5 - x} d x = 2 (- 2 + 5 ln (3))

\int_{2}^{4} \frac{5}{5 - x} d x = 5 ln (3)

= 2 (- 2 + 5 ln (3)) + 5 ln (3)

Simplify

2 (- 2 + 5 ln (3)) + 5 ln (3) : - 4 + 15 ln (3)

= - 4 + 15 ln (3)

What is the integral from 2 to 4 of (2x+5)/(5-x) ?
The integral from 2 to 4 of (2x+5)/(5-x) is -4+15ln(3)