What is the integral of (4x^2+x+5)/((x+1)(x^2+1)) ?

The integral of (4x^2+x+5)/((x+1)(x^2+1)) is 4ln|x+1|+arctan(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4x^2+x+5)/((x+1)(x^2+1))

\int \frac{4 x ^{2} + x + 5}{( x + 1 ) ( x ^{2} + 1 )} d x

4 ln ∣ x + 1 ∣ + arctan (x) + C

Solution steps

\int \frac{4 x ^{2} + x + 5}{( x + 1 ) ( x ^{2} + 1 )} d x

Expand

\frac{4 x ^{2} + x + 5}{( x + 1 ) ( x ^{2} + 1 )} : \frac{4 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} + \frac{x}{( x + 1 ) ( x ^{2} + 1 )} + \frac{5}{( x + 1 ) ( x ^{2} + 1 )}

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

\int \frac{4 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} d x = 4 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x)))

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

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

= 4 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x))) - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + 5 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)))

Simplify

4 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 - arctan (x))) - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (\frac{1}{2} ln x^{2} + 1 + arctan (x)) + 5 (\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x))) : 4 ln ∣ x + 1 ∣ + arctan (x)

= 4 ln ∣ x + 1 ∣ + arctan (x)

Add a constant to the solution

= 4 ln ∣ x + 1 ∣ + arctan (x) + C

\int (8 x^{5} - 3 x^{2} - 2 x + 5) d x

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

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

\int \frac{x + 4}{x ^{2} + 4 x - 12} d x

\int cot^{\frac{1}{2}} (3 x) csc^{4} (3 x) d x

What is the integral of (4x^2+x+5)/((x+1)(x^2+1)) ?
The integral of (4x^2+x+5)/((x+1)(x^2+1)) is 4ln|x+1|+arctan(x)+C