What is the integral of (a+b/(x-a))^2 ?

The integral of (a+b/(x-a))^2 is a^2x+2abln|x-a|-(b^2)/(x-a)+C

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integral of (a+b/(x-a))^2

\int (a + \frac{b}{x - a})^{2} d x

a^{2} x + 2 ab ln ∣ x - a ∣ - \frac{b ^{2}}{x - a} + C

Solution steps

\int (a + \frac{b}{x - a})^{2} d x

Expand

(a + \frac{b}{x - a})^{2} : a^{2} + \frac{2 ab}{x - a} + \frac{b ^{2}}{x ^{2} - 2 a x + a ^{2}}

= \int a^{2} + \frac{2 ab}{x - a} + \frac{b ^{2}}{x ^{2} - 2 a x + a ^{2}} d x

Factor

\frac{b ^{2}}{x ^{2} - 2 a x + a ^{2}}

= \int a^{2} + \frac{2 ab}{x - a} + \frac{b ^{2}}{( x - a ) ^{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 a^{2} d x + \int \frac{2 ab}{x - a} d x + \int \frac{b ^{2}}{( x - a ) ^{2}} d x

\int a^{2} d x = a^{2} x

\int \frac{2 ab}{x - a} d x = 2 ab ln ∣ x - a ∣

\int \frac{b ^{2}}{( x - a ) ^{2}} d x = - \frac{b ^{2}}{x - a}

= a^{2} x + 2 ab ln ∣ x - a ∣ - \frac{b ^{2}}{x - a}

Add a constant to the solution

= a^{2} x + 2 ab ln ∣ x - a ∣ - \frac{b ^{2}}{x - a} + C

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

f (x) = \frac{1}{3 x ^{2}}

\int 7 sin^{2} (x) cos^{2} (x) d x

\int (2 sec^{2} (x) - 5 cos (x)) d x

lo g_{10} (x + 1)

What is the integral of (a+b/(x-a))^2 ?
The integral of (a+b/(x-a))^2 is a^2x+2abln|x-a|-(b^2)/(x-a)+C