What is the integral of (48x^2)/(x^4-74x^2+1225) ?

The integral of (48x^2)/(x^4-74x^2+1225) is 5ln|x+5|-5ln|x-5|-7ln|x+7|+7ln|x-7|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (48x^2)/(x^4-74x^2+1225)

\int \frac{48 x ^{2}}{x ^{4} - 74 x ^{2} + 1225} d x

5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{48 x ^{2}}{x ^{4} - 74 x ^{2} + 1225} d x

Take the constant out:

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

= 48 \cdot \int \frac{x ^{2}}{x ^{4} - 74 x ^{2} + 1225} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{4} - 74 x ^{2} + 1225} : \frac{5}{48 ( x + 5 )} - \frac{5}{48 ( x - 5 )} - \frac{7}{48 ( x + 7 )} + \frac{7}{48 ( x - 7 )}

= 48 \cdot \int \frac{5}{48 ( x + 5 )} - \frac{5}{48 ( x - 5 )} - \frac{7}{48 ( x + 7 )} + \frac{7}{48 ( 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

= 48 (\int \frac{5}{48 ( x + 5 )} d x - \int \frac{5}{48 ( x - 5 )} d x - \int \frac{7}{48 ( x + 7 )} d x + \int \frac{7}{48 ( x - 7 )} d x)

\int \frac{5}{48 ( x + 5 )} d x = \frac{5}{48} ln ∣ x + 5 ∣

\int \frac{5}{48 ( x - 5 )} d x = \frac{5}{48} ln ∣ x - 5 ∣

\int \frac{7}{48 ( x + 7 )} d x = \frac{7}{48} ln ∣ x + 7 ∣

\int \frac{7}{48 ( x - 7 )} d x = \frac{7}{48} ln ∣ x - 7 ∣

= 48 (\frac{5}{48} ln ∣ x + 5 ∣ - \frac{5}{48} ln ∣ x - 5 ∣ - \frac{7}{48} ln ∣ x + 7 ∣ + \frac{7}{48} ln ∣ x - 7 ∣)

Simplify

48 (\frac{5}{48} ln ∣ x + 5 ∣ - \frac{5}{48} ln ∣ x - 5 ∣ - \frac{7}{48} ln ∣ x + 7 ∣ + \frac{7}{48} ln ∣ x - 7 ∣) : 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

Add a constant to the solution

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

What is the integral of (48x^2)/(x^4-74x^2+1225) ?
The integral of (48x^2)/(x^4-74x^2+1225) is 5ln|x+5|-5ln|x-5|-7ln|x+7|+7ln|x-7|+C