What is the integral of (y^2)/((y-x)^2) ?

The integral of (y^2)/((y-x)^2) is 2x(-y/(-x+y)+ln|-x+y|)+(x^2)/(-x+y)+y+C

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integral of (y^2)/((y-x)^2)

integral

\frac{y ^{2}}{( y - x ) ^{2}}

2 x (- \frac{y}{- x + y} + ln ∣ - x + y ∣) + \frac{x ^{2}}{- x + y} + y + C

Solution steps

\int \frac{y ^{2}}{( y - x ) ^{2}} d y

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

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

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{2 x y - x ^{2}}{( - x + y ) ^{2}} d y + \int 1 d y

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

\int 1 d y = y

= 2 x (- \frac{y}{- x + y} + ln ∣ - x + y ∣) + \frac{x ^{2}}{- x + y} + y

Add a constant to the solution

= 2 x (- \frac{y}{- x + y} + ln ∣ - x + y ∣) + \frac{x ^{2}}{- x + y} + y + C

\frac{x ^{2} + 2}{x + 2}

\frac{( x + 3 )}{( x ^{2} + 4 x + 7 )}

2 x^{2} + 1

e^{- 1} f (x)

tan^{5} (x)

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