What is the integral of (8x-13)/(x^2+2x+5) ?

The integral of (8x-13)/(x^2+2x+5) is -21/2 arctan(1/2 (x+1))+4ln|x^2+2x+5|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (8x-13)/(x^2+2x+5)

\int \frac{8 x - 13}{x ^{2} + 2 x + 5} d x

- \frac{21}{2} arctan (\frac{1}{2} (x + 1)) + 4 ln x^{2} + 2 x + 5 + C

Solution steps

\int \frac{8 x - 13}{x ^{2} + 2 x + 5} d x

Apply Integration By Parts

: \frac{1}{2} arctan (\frac{1}{2} (x + 1)) (8 x - 13) - \int 4 arctan (\frac{1}{2} (x + 1)) d x

= \frac{1}{2} arctan (\frac{1}{2} (x + 1)) (8 x - 13) - \int 4 arctan (\frac{1}{2} (x + 1)) d x

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

= \frac{1}{2} arctan (\frac{1}{2} (x + 1)) (8 x - 13) - 4 (x arctan (\frac{1}{2} (x + 1)) - ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1)))

Simplify

\frac{1}{2} arctan (\frac{1}{2} (x + 1)) (8 x - 13) - 4 (x arctan (\frac{1}{2} (x + 1)) - ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1))) : - \frac{21}{2} arctan (\frac{1}{2} (x + 1)) + 4 ln x^{2} + 2 x + 5

= - \frac{21}{2} arctan (\frac{1}{2} (x + 1)) + 4 ln x^{2} + 2 x + 5

Add a constant to the solution

= - \frac{21}{2} arctan (\frac{1}{2} (x + 1)) + 4 ln x^{2} + 2 x + 5 + C

\int \frac{1}{x ( 4 x - 1 )} d x

\int - \frac{6}{x ^{7}} d x

\int sec (x) sin (2 x) d x

\int 45 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{12} d x

\int 2 cos (6 x) d x

What is the integral of (8x-13)/(x^2+2x+5) ?
The integral of (8x-13)/(x^2+2x+5) is -21/2 arctan(1/2 (x+1))+4ln|x^2+2x+5|+C