What is the integral of (2x+3)/(x^2(x+5)(x^2+1)^2) ?

The integral of (2x+3)/(x^2(x+5)(x^2+1)^2) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x+3)/(x^2(x+5)(x^2+1)^2)

\int \frac{2 x + 3}{x ^{2} ( x + 5 ) ( x ^{2} + 1 ) ^{2}} d x

\frac{7}{25} ln ∣ x ∣ - \frac{157}{169} arctan (x) - \frac{17}{52} \cdot \frac{x}{x ^{2} + 1} - \frac{7}{16900} ln ∣ x + 5 ∣ - \frac{189}{1352} ln x^{2} + 1 - \frac{3}{5 x} + \frac{7}{52 ( x ^{2} + 1 )} + C

Solution steps

\int \frac{2 x + 3}{x ^{2} ( x + 5 ) ( x ^{2} + 1 ) ^{2}} d x

Take the partial fraction of

\frac{2 x + 3}{x ^{2} ( x + 5 ) ( x ^{2} + 1 ) ^{2}} : \frac{7}{25 x} + \frac{3}{5 x ^{2}} - \frac{7}{16900 ( x + 5 )} + \frac{- 189 x - 407}{676 ( x ^{2} + 1 )} + \frac{- 7 x - 17}{26 ( x ^{2} + 1 ) ^{2}}

= \int \frac{7}{25 x} + \frac{3}{5 x ^{2}} - \frac{7}{16900 ( x + 5 )} + \frac{- 189 x - 407}{676 ( x ^{2} + 1 )} + \frac{- 7 x - 17}{26 ( x ^{2} + 1 ) ^{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{7}{25 x} d x + \int \frac{3}{5 x ^{2}} d x - \int \frac{7}{16900 ( x + 5 )} d x + \int \frac{- 189 x - 407}{676 ( x ^{2} + 1 )} d x + \int \frac{- 7 x - 17}{26 ( x ^{2} + 1 ) ^{2}} d x

\int \frac{7}{25 x} d x = \frac{7}{25} ln ∣ x ∣

\int \frac{3}{5 x ^{2}} d x = - \frac{3}{5 x}

\int \frac{7}{16900 ( x + 5 )} d x = \frac{7}{16900} ln ∣ x + 5 ∣

\int \frac{- 189 x - 407}{676 ( x ^{2} + 1 )} d x = \frac{1}{676} (- \frac{189}{2} ln x^{2} + 1 - 407 arctan (x))

\int \frac{- 7 x - 17}{26 ( x ^{2} + 1 ) ^{2}} d x = \frac{1}{26} (\frac{7}{2 ( x ^{2} + 1 )} - \frac{17}{2} (arctan (x) + \frac{x}{1 + x ^{2}}))

= \frac{7}{25} ln ∣ x ∣ - \frac{3}{5 x} - \frac{7}{16900} ln ∣ x + 5 ∣ + \frac{1}{676} (- \frac{189}{2} ln x^{2} + 1 - 407 arctan (x)) + \frac{1}{26} (\frac{7}{2 ( x ^{2} + 1 )} - \frac{17}{2} (arctan (x) + \frac{x}{1 + x ^{2}}))

Simplify

\frac{7}{25} ln ∣ x ∣ - \frac{3}{5 x} - \frac{7}{16900} ln ∣ x + 5 ∣ + \frac{1}{676} (- \frac{189}{2} ln x^{2} + 1 - 407 arctan (x)) + \frac{1}{26} (\frac{7}{2 ( x ^{2} + 1 )} - \frac{17}{2} (arctan (x) + \frac{x}{1 + x ^{2}})) : \frac{7}{25} ln ∣ x ∣ - \frac{157}{169} arctan (x) - \frac{17}{52} \cdot \frac{x}{x ^{2} + 1} - \frac{7}{16900} ln ∣ x + 5 ∣ - \frac{189}{1352} ln x^{2} + 1 - \frac{3}{5 x} + \frac{7}{52 ( x ^{2} + 1 )}

= \frac{7}{25} ln ∣ x ∣ - \frac{157}{169} arctan (x) - \frac{17}{52} \cdot \frac{x}{x ^{2} + 1} - \frac{7}{16900} ln ∣ x + 5 ∣ - \frac{189}{1352} ln x^{2} + 1 - \frac{3}{5 x} + \frac{7}{52 ( x ^{2} + 1 )}

Add a constant to the solution

= \frac{7}{25} ln ∣ x ∣ - \frac{157}{169} arctan (x) - \frac{17}{52} \cdot \frac{x}{x ^{2} + 1} - \frac{7}{16900} ln ∣ x + 5 ∣ - \frac{189}{1352} ln x^{2} + 1 - \frac{3}{5 x} + \frac{7}{52 ( x ^{2} + 1 )} + C

\frac{\partial}{\partial x} (sin (y - x))

e^{x^{3}}

\int \frac{2 x - 5}{x ^{2} - 10 x + 50} d x

6 t^{3}

x^{2} y^{^{'}} + 2 x y = 17 cos^{2} (x)

What is the integral of (2x+3)/(x^2(x+5)(x^2+1)^2) ?
The integral of (2x+3)/(x^2(x+5)(x^2+1)^2) is