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

The integral of 1/((x+1)^3(x-2)) is -1/27 ln|x+1|+1/(9(x+1))+1/(6(x+1)^2)+1/27 ln|x-2|+C

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

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

- \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ^{3} ( u - 3 )} d u

= \int \frac{1}{u ^{3} ( u - 3 )} d u

Take the partial fraction of

\frac{1}{u ^{3} ( u - 3 )} : - \frac{1}{27 u} - \frac{1}{9 u ^{2}} - \frac{1}{3 u ^{3}} + \frac{1}{27 ( u - 3 )}

= \int - \frac{1}{27 u} - \frac{1}{9 u ^{2}} - \frac{1}{3 u ^{3}} + \frac{1}{27 ( u - 3 )} d u

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{1}{27 u} d u - \int \frac{1}{9 u ^{2}} d u - \int \frac{1}{3 u ^{3}} d u + \int \frac{1}{27 ( u - 3 )} d u

\int \frac{1}{27 u} d u = \frac{1}{27} ln ∣ u ∣

\int \frac{1}{9 u ^{2}} d u = - \frac{1}{9 u}

\int \frac{1}{3 u ^{3}} d u = - \frac{1}{6 u ^{2}}

\int \frac{1}{27 ( u - 3 )} d u = \frac{1}{27} ln ∣ u - 3 ∣

= - \frac{1}{27} ln ∣ u ∣ - (- \frac{1}{9 u}) - (- \frac{1}{6 u ^{2}}) + \frac{1}{27} ln ∣ u - 3 ∣

Substitute back

u = x + 1

= - \frac{1}{27} ln ∣ x + 1 ∣ - (- \frac{1}{9 ( x + 1 )}) - (- \frac{1}{6 ( x + 1 ) ^{2}}) + \frac{1}{27} ln ∣ x + 1 - 3 ∣

Simplify

- \frac{1}{27} ln ∣ x + 1 ∣ - (- \frac{1}{9 ( x + 1 )}) - (- \frac{1}{6 ( x + 1 ) ^{2}}) + \frac{1}{27} ln ∣ x + 1 - 3 ∣ : - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣

= - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣

Add a constant to the solution

= - \frac{1}{27} ln ∣ x + 1 ∣ + \frac{1}{9 ( x + 1 )} + \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{27} ln ∣ x - 2 ∣ + C

\int \frac{3}{3 x - 2} d x

\int cos (u) d u

\int \frac{x ^{2} - 1}{x - 2} d x

\int 1 - x - x^{2} d x

\int cos (a x) e^{s i n (a x)} d x

What is the integral of 1/((x+1)^3(x-2)) ?
The integral of 1/((x+1)^3(x-2)) is -1/27 ln|x+1|+1/(9(x+1))+1/(6(x+1)^2)+1/27 ln|x-2|+C