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

The integral of (5x^3+2x)/(x-1) is 17x+7ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-17+C

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

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

17 x + 7 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 17 + C

Solution steps

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

Expand

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

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

Simplify

5 (\frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣) + 2 (x - 1 + ln ∣ x - 1 ∣) : 17 x + 7 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 17

= 17 x + 7 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 17

Add a constant to the solution

= 17 x + 7 ln ∣ x - 1 ∣ + \frac{5}{3} (x - 1)^{3} + \frac{15}{2} (x - 1)^{2} - 17 + C

\frac{d}{d x} (2 + 4 cos (x^{3}) + 3 sin (x^{3}))

f (x) = \frac{1}{5} x^{2} + \frac{200000}{x}

(1 + x^{2}) y^{^{''}} + y^{^{'}} x + a x = 0

f (t) = 3 cos (t)

What is the integral of (5x^3+2x)/(x-1) ?
The integral of (5x^3+2x)/(x-1) is 17x+7ln|x-1|+5/3 (x-1)^3+15/2 (x-1)^2-17+C