What is the integral of (3x^{11})/(x^2-x-6) ?

The integral of (3x^{11})/(x^2-x-6) is

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integral of (3x^{11})/(x^2-x-6)

\int \frac{3 x ^{11}}{x ^{2} - x - 6} d x

\frac{3 x ^{10}}{10} + \frac{x ^{9}}{3} + \frac{21 x ^{8}}{8} + \frac{39 x ^{7}}{7} + \frac{55 x ^{6}}{2} + \frac{399 x ^{5}}{5} + \frac{1389 x ^{4}}{4} + 1261 x^{3} + \frac{12117 x ^{2}}{2} + 34815 x + \frac{6144}{5} ln ∣ x + 2 ∣ + \frac{531441}{5} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{3 x ^{11}}{x ^{2} - x - 6} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{x ^{11}}{x ^{2} - x - 6} d x

Long division

\frac{x ^{11}}{x ^{2} - x - 6} : x^{9} + x^{8} + 7 x^{7} + 13 x^{6} + 55 x^{5} + 133 x^{4} + 463 x^{3} + 1261 x^{2} + 4039 x + 11605 + \frac{35839 x + 69630}{x ^{2} - x - 6}

= 3 \cdot \int x^{9} + x^{8} + 7 x^{7} + 13 x^{6} + 55 x^{5} + 133 x^{4} + 463 x^{3} + 1261 x^{2} + 4039 x + 11605 + \frac{35839 x + 69630}{x ^{2} - x - 6} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int x^{9} d x + \int x^{8} d x + \int 7 x^{7} d x + \int 13 x^{6} d x + \int 55 x^{5} d x + \int 133 x^{4} d x + \int 463 x^{3} d x + \int 1261 x^{2} d x + \int 4039 x d x + \int 11605 d x + \int \frac{35839 x + 69630}{x ^{2} - x - 6} d x)

\int x^{9} d x = \frac{x ^{10}}{10}

\int x^{8} d x = \frac{x ^{9}}{9}

\int 7 x^{7} d x = \frac{7 x ^{8}}{8}

\int 13 x^{6} d x = \frac{13 x ^{7}}{7}

\int 55 x^{5} d x = \frac{55 x ^{6}}{6}

\int 133 x^{4} d x = \frac{133 x ^{5}}{5}

\int 463 x^{3} d x = \frac{463 x ^{4}}{4}

\int 1261 x^{2} d x = \frac{1261 x ^{3}}{3}

\int 4039 x d x = \frac{4039 x ^{2}}{2}

\int 11605 d x = 11605 x

\int \frac{35839 x + 69630}{x ^{2} - x - 6} d x = \frac{2048}{5} ln ∣ x + 2 ∣ + \frac{177147}{5} ln ∣ x - 3 ∣

= 3 (\frac{x ^{10}}{10} + \frac{x ^{9}}{9} + \frac{7 x ^{8}}{8} + \frac{13 x ^{7}}{7} + \frac{55 x ^{6}}{6} + \frac{133 x ^{5}}{5} + \frac{463 x ^{4}}{4} + \frac{1261 x ^{3}}{3} + \frac{4039 x ^{2}}{2} + 11605 x + \frac{2048}{5} ln ∣ x + 2 ∣ + \frac{177147}{5} ln ∣ x - 3 ∣)

Simplify

3 (\frac{x ^{10}}{10} + \frac{x ^{9}}{9} + \frac{7 x ^{8}}{8} + \frac{13 x ^{7}}{7} + \frac{55 x ^{6}}{6} + \frac{133 x ^{5}}{5} + \frac{463 x ^{4}}{4} + \frac{1261 x ^{3}}{3} + \frac{4039 x ^{2}}{2} + 11605 x + \frac{2048}{5} ln ∣ x + 2 ∣ + \frac{177147}{5} ln ∣ x - 3 ∣) : \frac{3 x ^{10}}{10} + \frac{x ^{9}}{3} + \frac{21 x ^{8}}{8} + \frac{39 x ^{7}}{7} + \frac{55 x ^{6}}{2} + \frac{399 x ^{5}}{5} + \frac{1389 x ^{4}}{4} + 1261 x^{3} + \frac{12117 x ^{2}}{2} + 34815 x + \frac{6144}{5} ln ∣ x + 2 ∣ + \frac{531441}{5} ln ∣ x - 3 ∣

= \frac{3 x ^{10}}{10} + \frac{x ^{9}}{3} + \frac{21 x ^{8}}{8} + \frac{39 x ^{7}}{7} + \frac{55 x ^{6}}{2} + \frac{399 x ^{5}}{5} + \frac{1389 x ^{4}}{4} + 1261 x^{3} + \frac{12117 x ^{2}}{2} + 34815 x + \frac{6144}{5} ln ∣ x + 2 ∣ + \frac{531441}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{3 x ^{10}}{10} + \frac{x ^{9}}{3} + \frac{21 x ^{8}}{8} + \frac{39 x ^{7}}{7} + \frac{55 x ^{6}}{2} + \frac{399 x ^{5}}{5} + \frac{1389 x ^{4}}{4} + 1261 x^{3} + \frac{12117 x ^{2}}{2} + 34815 x + \frac{6144}{5} ln ∣ x + 2 ∣ + \frac{531441}{5} ln ∣ x - 3 ∣ + C

\int x \cdot x + 6 d x

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

\int_{- \infty}^{0} \frac{1}{4 - 9 x} d x

f (x) = x^{2} - 4

What is the integral of (3x^{11})/(x^2-x-6) ?
The integral of (3x^{11})/(x^2-x-6) is