What is the integral of (5x^3+3x)/(x-1) ?

The integral of (5x^3+3x)/(x-1) is 18x+8ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-18+C

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integral of (5x^3+3x)/(x-1)

\int \frac{5 x ^{3} + 3 x}{x - 1} d x

18 x + 8 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 18 + C

Solution steps

\int \frac{5 x ^{3} + 3 x}{x - 1} d x

Expand

\frac{5 x ^{3} + 3 x}{x - 1} : \frac{5 x ^{3}}{x - 1} + \frac{3 x}{x - 1}

Apply the Sum Rule:

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

= \int \frac{5 x ^{3}}{x - 1} d x + \int \frac{3 x}{x - 1} d x

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

\int \frac{3 x}{x - 1} d x = 3 (x - 1 + ln ∣ x - 1 ∣)

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

Simplify

5 (\frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣) + 3 (x - 1 + ln ∣ x - 1 ∣) : 18 x + 8 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 18

= 18 x + 8 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 18

Add a constant to the solution

= 18 x + 8 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 18 + C

\int (x^{5} + 1) d x

y^{^{''}} + 8 y^{^{'}} - 9 y = e^{3 x}

\frac{d}{d x} (x (x - 21)^{2})

\frac{( e ^{- 2 s} )}{s + 1}

\frac{d}{d x} (3 x - 2)

What is the integral of (5x^3+3x)/(x-1) ?
The integral of (5x^3+3x)/(x-1) is 18x+8ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-18+C