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

The integral of (5x^3+5x)/(x-1) is 20x+10ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-20+C

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

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

20 x + 10 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 20 + C

Solution steps

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

Expand

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

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

Simplify

5 (\frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣) + 5 (x - 1 + ln ∣ x - 1 ∣) : 20 x + 10 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 20

= 20 x + 10 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 20

Add a constant to the solution

= 20 x + 10 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 20 + C

y^{^{'}} + y = e^{t} + 11

\frac{d}{d x} (12 x^{3} - 36 x^{2})

\frac{\partial}{\partial x} (e^{- 5 x^{2} - 4 y^{2}})

d er i v a t i v e f (x) = ln (\frac{x}{x ^{2} + 1})

\int 5^{t} sin (5^{t}) d t

What is the integral of (5x^3+5x)/(x-1) ?
The integral of (5x^3+5x)/(x-1) is 20x+10ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-20+C