What is the integral from 1 to 3 of (5x+3)/(x^2+2x) ?

The integral from 1 to 3 of (5x+3)/(x^2+2x) is (-4ln(3)+7ln(5))/2

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integral from 1 to 3 of (5x+3)/(x^2+2x)

\int_{1}^{3} \frac{5 x + 3}{x ^{2} + 2 x} d x

\frac{- 4 ln ( 3 ) + 7 ln ( 5 )}{2}

Decimal

3.43580 \dots

Solution steps

\int_{1}^{3} \frac{5 x + 3}{x ^{2} + 2 x} d x

Take the partial fraction of

\frac{5 x + 3}{x ^{2} + 2 x} : \frac{3}{2 x} + \frac{7}{2 ( x + 2 )}

= \int_{1}^{3} \frac{3}{2 x} + \frac{7}{2 ( x + 2 )} 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}^{3} \frac{3}{2 x} d x + \int_{1}^{3} \frac{7}{2 ( x + 2 )} d x

\int_{1}^{3} \frac{3}{2 x} d x = \frac{3}{2} ln (3)

\int_{1}^{3} \frac{7}{2 ( x + 2 )} d x = \frac{7}{2} (ln (5) - ln (3))

= \frac{3}{2} ln (3) + \frac{7}{2} (ln (5) - ln (3))

Simplify

\frac{3}{2} ln (3) + \frac{7}{2} (ln (5) - ln (3)) : \frac{- 4 ln ( 3 ) + 7 ln ( 5 )}{2}

= \frac{- 4 ln ( 3 ) + 7 ln ( 5 )}{2}

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What is the integral from 1 to 3 of (5x+3)/(x^2+2x) ?
The integral from 1 to 3 of (5x+3)/(x^2+2x) is (-4ln(3)+7ln(5))/2