What is the integral of (x^6)/(x^2-4) ?

The integral of (x^6)/(x^2-4) is 32((x^5)/(160)+(x^3)/(24)+x/2-1/2 ln| x/2+1|+1/2 ln| x/2-1|)+C

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integral of (x^6)/(x^2-4)

\int \frac{x ^{6}}{x ^{2} - 4} d x

32 (\frac{x ^{5}}{160} + \frac{x ^{3}}{24} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1) + C

Solution steps

\int \frac{x ^{6}}{x ^{2} - 4} d x

Apply Integral Substitution

= \int \frac{32 u ^{6}}{u ^{2} - 1} d u

Take the constant out:

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

= 32 \cdot \int \frac{u ^{6}}{u ^{2} - 1} d u

Long division

\frac{u ^{6}}{u ^{2} - 1} : u^{4} + u^{2} + 1 + \frac{1}{u ^{2} - 1}

= 32 \cdot \int u^{4} + u^{2} + 1 + \frac{1}{u ^{2} - 1} 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

= 32 (\int u^{4} d u + \int u^{2} d u + \int 1 d u + \int \frac{1}{u ^{2} - 1} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

\int \frac{1}{u ^{2} - 1} d u = - \frac{1}{2} ln ∣ u + 1 ∣ + \frac{1}{2} ln ∣ u - 1 ∣

= 32 (\frac{u ^{5}}{5} + \frac{u ^{3}}{3} + u - \frac{1}{2} ln ∣ u + 1 ∣ + \frac{1}{2} ln ∣ u - 1 ∣)

Substitute back

u = \frac{x}{2}

= 32 (\frac{( \frac{x}{2} ) ^{5}}{5} + \frac{( \frac{x}{2} ) ^{3}}{3} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1)

Simplify

32 (\frac{( \frac{x}{2} ) ^{5}}{5} + \frac{( \frac{x}{2} ) ^{3}}{3} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1) : 32 (\frac{x ^{5}}{160} + \frac{x ^{3}}{24} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1)

= 32 (\frac{x ^{5}}{160} + \frac{x ^{3}}{24} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1)

Add a constant to the solution

= 32 (\frac{x ^{5}}{160} + \frac{x ^{3}}{24} + \frac{x}{2} - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1) + C

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What is the integral of (x^6)/(x^2-4) ?
The integral of (x^6)/(x^2-4) is 32((x^5)/(160)+(x^3)/(24)+x/2-1/2 ln| x/2+1|+1/2 ln| x/2-1|)+C