What is the integral of (2x+3)/(x(x-4)^2(x^2+2)) ?

The integral of (2x+3)/(x(x-4)^2(x^2+2)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (2x+3)/(x(x-4)^2(x^2+2))

\int \frac{2 x + 3}{x ( x - 4 ) ^{2} ( x ^{2} + 2 )} d x

\frac{3}{32} ln ∣ x ∣ - \frac{203}{2592} ln ∣ x - 4 ∣ - \frac{11}{72 ( x - 4 )} + \frac{1}{324} (- \frac{5}{2} ln x^{2} + 2 + 262 arctan (\frac{x}{2})) + C

Solution steps

\int \frac{2 x + 3}{x ( x - 4 ) ^{2} ( x ^{2} + 2 )} d x

Take the partial fraction of

\frac{2 x + 3}{x ( x - 4 ) ^{2} ( x ^{2} + 2 )} : \frac{3}{32 x} - \frac{203}{2592 ( x - 4 )} + \frac{11}{72 ( x - 4 ) ^{2}} + \frac{- 5 x + 52}{324 ( x ^{2} + 2 )}

= \int \frac{3}{32 x} - \frac{203}{2592 ( x - 4 )} + \frac{11}{72 ( x - 4 ) ^{2}} + \frac{- 5 x + 52}{324 ( x ^{2} + 2 )} 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

= \int \frac{3}{32 x} d x - \int \frac{203}{2592 ( x - 4 )} d x + \int \frac{11}{72 ( x - 4 ) ^{2}} d x + \int \frac{- 5 x + 52}{324 ( x ^{2} + 2 )} d x

\int \frac{3}{32 x} d x = \frac{3}{32} ln ∣ x ∣

\int \frac{203}{2592 ( x - 4 )} d x = \frac{203}{2592} ln ∣ x - 4 ∣

\int \frac{11}{72 ( x - 4 ) ^{2}} d x = - \frac{11}{72 ( x - 4 )}

\int \frac{- 5 x + 52}{324 ( x ^{2} + 2 )} d x = \frac{1}{324} (- \frac{5}{2} ln x^{2} + 2 + 262 arctan (\frac{x}{2}))

= \frac{3}{32} ln ∣ x ∣ - \frac{203}{2592} ln ∣ x - 4 ∣ - \frac{11}{72 ( x - 4 )} + \frac{1}{324} (- \frac{5}{2} ln x^{2} + 2 + 262 arctan (\frac{x}{2}))

Add a constant to the solution

= \frac{3}{32} ln ∣ x ∣ - \frac{203}{2592} ln ∣ x - 4 ∣ - \frac{11}{72 ( x - 4 )} + \frac{1}{324} (- \frac{5}{2} ln x^{2} + 2 + 262 arctan (\frac{x}{2})) + C

What is the integral of (2x+3)/(x(x-4)^2(x^2+2)) ?
The integral of (2x+3)/(x(x-4)^2(x^2+2)) is