What is the integral of (66x^2)/(x^4-65x^2+784) ?

The integral of (66x^2)/(x^4-65x^2+784) is 4ln|x+4|-4ln|x-4|-7ln|x+7|+7ln|x-7|+C

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integral of (66x^2)/(x^4-65x^2+784)

\int \frac{66 x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{66 x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

Take the constant out:

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

= 66 \cdot \int \frac{x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{4} - 65 x ^{2} + 784} : \frac{2}{33 ( x + 4 )} - \frac{2}{33 ( x - 4 )} - \frac{7}{66 ( x + 7 )} + \frac{7}{66 ( x - 7 )}

= 66 \cdot \int \frac{2}{33 ( x + 4 )} - \frac{2}{33 ( x - 4 )} - \frac{7}{66 ( x + 7 )} + \frac{7}{66 ( x - 7 )} 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

= 66 (\int \frac{2}{33 ( x + 4 )} d x - \int \frac{2}{33 ( x - 4 )} d x - \int \frac{7}{66 ( x + 7 )} d x + \int \frac{7}{66 ( x - 7 )} d x)

\int \frac{2}{33 ( x + 4 )} d x = \frac{2}{33} ln ∣ x + 4 ∣

\int \frac{2}{33 ( x - 4 )} d x = \frac{2}{33} ln ∣ x - 4 ∣

\int \frac{7}{66 ( x + 7 )} d x = \frac{7}{66} ln ∣ x + 7 ∣

\int \frac{7}{66 ( x - 7 )} d x = \frac{7}{66} ln ∣ x - 7 ∣

= 66 (\frac{2}{33} ln ∣ x + 4 ∣ - \frac{2}{33} ln ∣ x - 4 ∣ - \frac{7}{66} ln ∣ x + 7 ∣ + \frac{7}{66} ln ∣ x - 7 ∣)

Simplify

66 (\frac{2}{33} ln ∣ x + 4 ∣ - \frac{2}{33} ln ∣ x - 4 ∣ - \frac{7}{66} ln ∣ x + 7 ∣ + \frac{7}{66} ln ∣ x - 7 ∣) : 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

Add a constant to the solution

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

\frac{\partial}{\partial x} (y sin (x y) - 2 x y)

\frac{d}{d x} (x^{2} - 25 I n (x^{2}))

\int 2 y^{3} d y

\frac{1}{2} (2 (7.5) + 1)^{- \frac{1}{2}}

2 - x^{2}, 6 - 5 x

What is the integral of (66x^2)/(x^4-65x^2+784) ?
The integral of (66x^2)/(x^4-65x^2+784) is 4ln|x+4|-4ln|x-4|-7ln|x+7|+7ln|x-7|+C