What is the integral of (6x+5)/((3x^2+5x)^6) ?

The integral of (6x+5)/((3x^2+5x)^6) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (6x+5)/((3x^2+5x)^6)

\int \frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} d x

- \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}} + C

Solution steps

\int \frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} d x

Take the partial fraction of

\frac{6 x + 5}{( 3 x ^{2} + 5 x ) ^{6}} : \frac{1134}{1953125 x ^{2}} - \frac{378}{390625 x ^{3}} + \frac{81}{78125 x ^{4}} - \frac{12}{15625 x ^{5}} + \frac{1}{3125 x ^{6}} - \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} - \frac{10206}{390625 ( 3 x + 5 ) ^{3}} - \frac{6561}{78125 ( 3 x + 5 ) ^{4}} - \frac{2916}{15625 ( 3 x + 5 ) ^{5}} - \frac{729}{3125 ( 3 x + 5 ) ^{6}}

= \int \frac{1134}{1953125 x ^{2}} - \frac{378}{390625 x ^{3}} + \frac{81}{78125 x ^{4}} - \frac{12}{15625 x ^{5}} + \frac{1}{3125 x ^{6}} - \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} - \frac{10206}{390625 ( 3 x + 5 ) ^{3}} - \frac{6561}{78125 ( 3 x + 5 ) ^{4}} - \frac{2916}{15625 ( 3 x + 5 ) ^{5}} - \frac{729}{3125 ( 3 x + 5 ) ^{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

= \int \frac{1134}{1953125 x ^{2}} d x - \int \frac{378}{390625 x ^{3}} d x + \int \frac{81}{78125 x ^{4}} d x - \int \frac{12}{15625 x ^{5}} d x + \int \frac{1}{3125 x ^{6}} d x - \int \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} d x - \int \frac{10206}{390625 ( 3 x + 5 ) ^{3}} d x - \int \frac{6561}{78125 ( 3 x + 5 ) ^{4}} d x - \int \frac{2916}{15625 ( 3 x + 5 ) ^{5}} d x - \int \frac{729}{3125 ( 3 x + 5 ) ^{6}} d x

\int \frac{1134}{1953125 x ^{2}} d x = - \frac{1134}{1953125 x}

\int \frac{378}{390625 x ^{3}} d x = - \frac{189}{390625 x ^{2}}

\int \frac{81}{78125 x ^{4}} d x = - \frac{27}{78125 x ^{3}}

\int \frac{12}{15625 x ^{5}} d x = - \frac{3}{15625 x ^{4}}

\int \frac{1}{3125 x ^{6}} d x = - \frac{1}{15625 x ^{5}}

\int \frac{10206}{1953125 ( 3 x + 5 ) ^{2}} d x = - \frac{3402}{1953125 ( 3 x + 5 )}

\int \frac{10206}{390625 ( 3 x + 5 ) ^{3}} d x = - \frac{1701}{390625 ( 3 x + 5 ) ^{2}}

\int \frac{6561}{78125 ( 3 x + 5 ) ^{4}} d x = - \frac{729}{78125 ( 3 x + 5 ) ^{3}}

\int \frac{2916}{15625 ( 3 x + 5 ) ^{5}} d x = - \frac{243}{15625 ( 3 x + 5 ) ^{4}}

\int \frac{729}{3125 ( 3 x + 5 ) ^{6}} d x = - \frac{243}{15625 ( 3 x + 5 ) ^{5}}

= - \frac{1134}{1953125 x} - (- \frac{189}{390625 x ^{2}}) - \frac{27}{78125 x ^{3}} - (- \frac{3}{15625 x ^{4}}) - \frac{1}{15625 x ^{5}} - (- \frac{3402}{1953125 ( 3 x + 5 )}) - (- \frac{1701}{390625 ( 3 x + 5 ) ^{2}}) - (- \frac{729}{78125 ( 3 x + 5 ) ^{3}}) - (- \frac{243}{15625 ( 3 x + 5 ) ^{4}}) - (- \frac{243}{15625 ( 3 x + 5 ) ^{5}})

Simplify

= - \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}}

Add a constant to the solution

= - \frac{1134}{1953125 x} + \frac{189}{390625 x ^{2}} - \frac{27}{78125 x ^{3}} + \frac{3}{15625 x ^{4}} - \frac{1}{15625 x ^{5}} + \frac{3402}{1953125 ( 3 x + 5 )} + \frac{1701}{390625 ( 3 x + 5 ) ^{2}} + \frac{729}{78125 ( 3 x + 5 ) ^{3}} + \frac{243}{15625 ( 3 x + 5 ) ^{4}} + \frac{243}{15625 ( 3 x + 5 ) ^{5}} + C

f (x) = \frac{( 1 - 4 x )}{3 + x}

\frac{d}{d x} (1 0^{9 - x^{2}})

t^{2} sin (2 t)

x y^{^{'}} = y + 4 x^{2} sin (x), y (π) = 0

(1 + x^{2}) \frac{d y}{d x} = \frac{x ^{1}}{y}

What is the integral of (6x+5)/((3x^2+5x)^6) ?
The integral of (6x+5)/((3x^2+5x)^6) is