What is the integral of (x+2)/(sqrt(x^2+2x-3)) ?

The integral of (x+2)/(sqrt(x^2+2x-3)) is sqrt(x^2+2x-3)+ln(1/2 |sqrt(x^2+2x-3)+x+1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+2)/(sqrt(x^2+2x-3))

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

x^{2} + 2 x - 3 + ln (\frac{1}{2} x^{2} + 2 x - 3 + x + 1) + C

Solution steps

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

Expand

\frac{x + 2}{x ^{2} + 2 x - 3} : \frac{x}{x ^{2} + 2 x - 3} + \frac{2}{x ^{2} + 2 x - 3}

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

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

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

= x^{2} + 2 x - 3 - ln (\frac{1}{2} x^{2} + 2 x - 3 + x + 1) + 2 ln (\frac{1}{2} x^{2} + 2 x - 3 + x + 1)

Simplify

= x^{2} + 2 x - 3 + ln (\frac{1}{2} x^{2} + 2 x - 3 + x + 1)

Add a constant to the solution

= x^{2} + 2 x - 3 + ln (\frac{1}{2} x^{2} + 2 x - 3 + x + 1) + C

\int \frac{( - 2 x + 6 )}{( x ^{2} + 2 x + 4 ) ^{2}} d x

\int \frac{x ^{2} + 10 x + 2}{x ^{3} + 15 x ^{2} + 6 x + 3} d x

\int (sec^{2} (π w) + e^{- 3 w}) d w

\int \frac{u ^{2}}{u ^{2} - 1} d u

\int \frac{x}{7 x ^{2} - 2} d x

What is the integral of (x+2)/(sqrt(x^2+2x-3)) ?
The integral of (x+2)/(sqrt(x^2+2x-3)) is sqrt(x^2+2x-3)+ln(1/2 |sqrt(x^2+2x-3)+x+1|)+C