What is the integral of x/((xy+1)^2) ?

The integral of x/((xy+1)^2) is -1/(1+xy)+C

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integral of x/((xy+1)^2)

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

- \frac{1}{1 + x y} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{x u ^{2}} d u

= x \cdot \int \frac{1}{x u ^{2}} d u

Take the constant out:

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

= x \frac{1}{x} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= x \frac{1}{x} (- \frac{1}{u})

Substitute back

u = 1 + x y

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

Simplify

x \frac{1}{x} (- \frac{1}{1 + x y}) : - \frac{1}{1 + x y}

= - \frac{1}{1 + x y}

Add a constant to the solution

= - \frac{1}{1 + x y} + C

2 y^{^{''}} + 20 y^{^{'}} + 50 y = 0

\frac{\partial}{\partial x} (2 x^{2} + y^{2})

\frac{d}{d x} (x^{25})

∣ x ∣ + 1

x \to 0 lim (4 \cdot x^{2})

What is the integral of x/((xy+1)^2) ?
The integral of x/((xy+1)^2) is -1/(1+xy)+C