What is the integral of (x^3)/(x^2+22x+121) ?

The integral of (x^3)/(x^2+22x+121) is 1/2 (x+11)^2-33(x+11)+363ln|x+11|+(1331)/(x+11)+C

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integral of (x^3)/(x^2+22x+121)

\int \frac{x ^{3}}{x ^{2} + 22 x + 121} d x

\frac{1}{2} (x + 11)^{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ + \frac{1331}{x + 11} + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 22 x + 121} d x

Complete the square

x^{2} + 22 x + 121 : (x + 11)^{2}

= \int \frac{x ^{3}}{( x + 11 ) ^{2}} d x

Apply u-substitution

= \int \frac{( u - 11 ) ^{3}}{u ^{2}} d u

Expand

\frac{( u - 11 ) ^{3}}{u ^{2}} : u - 33 + \frac{363}{u} - \frac{1331}{u ^{2}}

= \int u - 33 + \frac{363}{u} - \frac{1331}{u ^{2}} 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 u d u - \int 33 d u + \int \frac{363}{u} d u - \int \frac{1331}{u ^{2}} d u

\int u d u = \frac{u ^{2}}{2}

\int 33 d u = 33 u

\int \frac{363}{u} d u = 363 ln ∣ u ∣

\int \frac{1331}{u ^{2}} d u = - \frac{1331}{u}

= \frac{u ^{2}}{2} - 33 u + 363 ln ∣ u ∣ - (- \frac{1331}{u})

Substitute back

u = x + 11

= \frac{( x + 11 ) ^{2}}{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ - (- \frac{1331}{x + 11})

Simplify

\frac{( x + 11 ) ^{2}}{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ - (- \frac{1331}{x + 11}) : \frac{1}{2} (x + 11)^{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ + \frac{1331}{x + 11}

= \frac{1}{2} (x + 11)^{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ + \frac{1331}{x + 11}

Add a constant to the solution

= \frac{1}{2} (x + 11)^{2} - 33 (x + 11) + 363 ln ∣ x + 11 ∣ + \frac{1331}{x + 11} + C

What is the integral of (x^3)/(x^2+22x+121) ?
The integral of (x^3)/(x^2+22x+121) is 1/2 (x+11)^2-33(x+11)+363ln|x+11|+(1331)/(x+11)+C