What is the integral of (2x+16)/(x(x^2-25)) ?

The integral of (2x+16)/(x(x^2-25)) is -1/5 (ln| x/5+1|-ln| x/5-1|)+8(-2/25 ln(x)+1/25 ln|x^2-25|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x+16)/(x(x^2-25))

\int \frac{2 x + 16}{x ( x ^{2} - 25 )} d x

- \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1) + 8 (- \frac{2}{25} ln (x) + \frac{1}{25} ln x^{2} - 25) + C

Solution steps

\int \frac{2 x + 16}{x ( x ^{2} - 25 )} d x

Expand

\frac{2 x + 16}{x ( x ^{2} - 25 )} : \frac{2 x}{x ( x ^{2} - 25 )} + \frac{16}{x ( x ^{2} - 25 )}

Apply the Sum Rule:

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

= \int \frac{2 x}{x ( x ^{2} - 25 )} d x + \int \frac{16}{x ( x ^{2} - 25 )} d x

\int \frac{2 x}{x ( x ^{2} - 25 )} d x = - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1)

\int \frac{16}{x ( x ^{2} - 25 )} d x = 8 (- \frac{2}{25} ln (x) + \frac{1}{25} ln x^{2} - 25)

= - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1) + 8 (- \frac{2}{25} ln (x) + \frac{1}{25} ln x^{2} - 25)

Add a constant to the solution

= - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1) + 8 (- \frac{2}{25} ln (x) + \frac{1}{25} ln x^{2} - 25) + C

\int \frac{1}{x ^{\frac{1}{2}} + x ^{\frac{2}{3}}} d x

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

\int \frac{( y - 3 )}{- 24 + 14 y - y ^{2}} d y

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

\int \frac{x ^{2}}{x ^{2} - 7} d x

What is the integral of (2x+16)/(x(x^2-25)) ?
The integral of (2x+16)/(x(x^2-25)) is -1/5 (ln| x/5+1|-ln| x/5-1|)+8(-2/25 ln(x)+1/25 ln|x^2-25|)+C