What is the integral of (5x^3)/((x-2)(x+3)) ?

The integral of (5x^3)/((x-2)(x+3)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (5x^3)/((x-2)(x+3))

\int \frac{5 x ^{3}}{( x - 2 ) ( x + 3 )} d x

\frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5} + C

Solution steps

\int \frac{5 x ^{3}}{( x - 2 ) ( x + 3 )} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{3}}{( x - 2 ) ( x + 3 )} d x

Long division

\frac{x ^{3}}{( x - 2 ) ( x + 3 )} : x - 1 + \frac{7 x - 6}{x ^{2} + x - 6}

= 5 \cdot \int x - 1 + \frac{7 x - 6}{x ^{2} + x - 6} 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

= 5 (\int x d x - \int 1 d x + \int \frac{7 x - 6}{x ^{2} + x - 6} d x)

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

\int \frac{7 x - 6}{x ^{2} + x - 6} d x = 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= 5 (\frac{x ^{2}}{2} - x + 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}))

Simplify

5 (\frac{x ^{2}}{2} - x + 14 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) + \frac{6}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})) : \frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5}

= \frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5}

Add a constant to the solution

= \frac{5 x ^{2}}{2} - 5 x + \frac{35}{2} ln 4 x^{2} + 4 x - 24 + \frac{7}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + 6 ln \frac{2}{5} x + \frac{6}{5} - 6 ln \frac{2}{5} x - \frac{4}{5} + C

\frac{\partial}{\partial x} (ln (x + z) - yz)

\frac{d}{d t} (\frac{d x}{d t}) + \frac{d}{d t} (6 x) + 10 x = 0

y + 4 y^{3} = (y^{4} + 2 x) y^{^{'}}

\int x^{2} cos (k x) d x

y^{^{'}} - 2 x y = x, y (0) = 0

What is the integral of (5x^3)/((x-2)(x+3)) ?
The integral of (5x^3)/((x-2)(x+3)) is