What is the integral of (-x+5)/(34(x^2+9)) ?

The integral of (-x+5)/(34(x^2+9)) is 1/102 (-3/2 ln|(x^2)/9+1|+5arctan(x/3))+C

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integral of (-x+5)/(34(x^2+9))

\int \frac{- x + 5}{34 ( x ^{2} + 9 )} d x

\frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3})) + C

Solution steps

\int \frac{- x + 5}{34 ( x ^{2} + 9 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{34} \cdot \int \frac{- x + 5}{x ^{2} + 9} d x

Apply Integral Substitution

= \frac{1}{34} \cdot \int \frac{- 3 u + 5}{3 ( u ^{2} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{34} \cdot \frac{1}{3} \cdot \int \frac{- 3 u + 5}{u ^{2} + 1} d u

Expand

\frac{- 3 u + 5}{u ^{2} + 1} : - \frac{3 u}{u ^{2} + 1} + \frac{5}{u ^{2} + 1}

Apply the Sum Rule:

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

= \frac{1}{34} \cdot \frac{1}{3} (- \int \frac{3 u}{u ^{2} + 1} d u + \int \frac{5}{u ^{2} + 1} d u)

\int \frac{3 u}{u ^{2} + 1} d u = \frac{3}{2} ln u^{2} + 1

\int \frac{5}{u ^{2} + 1} d u = 5 arctan (u)

= \frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln u^{2} + 1 + 5 arctan (u))

Substitute back

u = \frac{x}{3}

= \frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 + 5 arctan (\frac{x}{3}))

Simplify

\frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 + 5 arctan (\frac{x}{3})) : \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3}))

= \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3}))

Add a constant to the solution

= \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3})) + C

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What is the integral of (-x+5)/(34(x^2+9)) ?
The integral of (-x+5)/(34(x^2+9)) is 1/102 (-3/2 ln|(x^2)/9+1|+5arctan(x/3))+C