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

The integral of (2x+1)/((3x-1)(2x+5)) is 1/51 (17ln|6x+15|-5(ln| 4/17 (3x-1)+2|-ln(4/17 |3x-1|)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{51} (17 ln ∣ 6 x + 15 ∣ - 5 (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣))) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u + 5}{3 u ( 2 u + 17 )} d u

= \int \frac{2 u + 5}{3 u ( 2 u + 17 )} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{2 u + 5}{u ( 2 u + 17 )} d u

Expand

\frac{2 u + 5}{u ( 2 u + 17 )} : \frac{2 u}{u ( 2 u + 17 )} + \frac{5}{u ( 2 u + 17 )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int \frac{2 u}{u ( 2 u + 17 )} d u + \int \frac{5}{u ( 2 u + 17 )} d u)

\int \frac{2 u}{u ( 2 u + 17 )} d u = ln ∣ 2 u + 17 ∣

\int \frac{5}{u ( 2 u + 17 )} d u = - \frac{5}{17} (ln \frac{4}{17} u + 2 - ln (\frac{4}{17} ∣ u ∣))

= \frac{1}{3} (ln ∣ 2 u + 17 ∣ - \frac{5}{17} (ln \frac{4}{17} u + 2 - ln (\frac{4}{17} ∣ u ∣)))

Substitute back

u = 3 x - 1

= \frac{1}{3} (ln ∣ 2 (3 x - 1) + 17 ∣ - \frac{5}{17} (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣)))

Simplify

\frac{1}{3} (ln ∣ 2 (3 x - 1) + 17 ∣ - \frac{5}{17} (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣))) : \frac{1}{51} (17 ln ∣ 6 x + 15 ∣ - 5 (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣)))

= \frac{1}{51} (17 ln ∣ 6 x + 15 ∣ - 5 (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣)))

Add a constant to the solution

= \frac{1}{51} (17 ln ∣ 6 x + 15 ∣ - 5 (ln \frac{4}{17} (3 x - 1) + 2 - ln (\frac{4}{17} ∣ 3 x - 1 ∣))) + C

\int 3 x^{2} \cdot cos (x) d x

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

\int \frac{3 ^{2 y} - 2 7 ^{y}}{9 ^{2 y}} d y

\int \frac{1}{3} sin^{3} (x) cos^{3} (x) d x

\int 2 ln (3 x + 1) d x

What is the integral of (2x+1)/((3x-1)(2x+5)) ?
The integral of (2x+1)/((3x-1)(2x+5)) is 1/51 (17ln|6x+15|-5(ln| 4/17 (3x-1)+2|-ln(4/17 |3x-1|)))+C