What is the integral of (7x^3+3x)/(x-1) ?

The integral of (7x^3+3x)/(x-1) is 24x+10ln|x-1|+7/3 (x-1)^3+21/2 (x-1)^2-24+C

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integral of (7x^3+3x)/(x-1)

\int \frac{7 x ^{3} + 3 x}{x - 1} d x

24 x + 10 ln ∣ x - 1 ∣ + \frac{7}{3} (x - 1)^{3} + \frac{21}{2} (x - 1)^{2} - 24 + C

Solution steps

\int \frac{7 x ^{3} + 3 x}{x - 1} d x

Expand

\frac{7 x ^{3} + 3 x}{x - 1} : \frac{7 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{7 x ^{3}}{x - 1} d x + \int \frac{3 x}{x - 1} d x

\int \frac{7 x ^{3}}{x - 1} d x = 7 (\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 ∣)

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

Simplify

7 (\frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣) + 3 (x - 1 + ln ∣ x - 1 ∣) : 24 x + 10 ln ∣ x - 1 ∣ + \frac{7}{3} (x - 1)^{3} + \frac{21}{2} (x - 1)^{2} - 24

= 24 x + 10 ln ∣ x - 1 ∣ + \frac{7}{3} (x - 1)^{3} + \frac{21}{2} (x - 1)^{2} - 24

Add a constant to the solution

= 24 x + 10 ln ∣ x - 1 ∣ + \frac{7}{3} (x - 1)^{3} + \frac{21}{2} (x - 1)^{2} - 24 + C

\int 3 e^{- 2 x} d x

\frac{2 x ^{4}}{cos ( x ) e ^{x}}

x \to 3 + lim (\frac{∣ x - 3 ∣}{x - 3})

\int \frac{1}{1 - sin ( x )} d x

x \to 0 lim (\frac{sin ( 2 ) ( x )}{x})

What is the integral of (7x^3+3x)/(x-1) ?
The integral of (7x^3+3x)/(x-1) is 24x+10ln|x-1|+7/3 (x-1)^3+21/2 (x-1)^2-24+C