What is the integral of (5x)/(1+xy) ?

The integral of (5x)/(1+xy) is 5ln|1+xy|+C

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integral of (5x)/(1+xy)

\int \frac{5 x}{1 + x y} d y

5 ln ∣ 1 + x y ∣ + C

Solution steps

\int \frac{5 x}{1 + x y} d y

Take the constant out:

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

= 5 x \cdot \int \frac{1}{1 + x y} d y

Apply u-substitution

= 5 x \cdot \int \frac{1}{xu} d u

Take the constant out:

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

= 5 x \frac{1}{x} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 x \frac{1}{x} ln ∣ u ∣

Substitute back

u = 1 + x y

= 5 x \frac{1}{x} ln ∣ 1 + x y ∣

Simplify

5 x \frac{1}{x} ln ∣ 1 + x y ∣ : 5 ln ∣ 1 + x y ∣

= 5 ln ∣ 1 + x y ∣

Add a constant to the solution

= 5 ln ∣ 1 + x y ∣ + C

What is the integral of (5x)/(1+xy) ?
The integral of (5x)/(1+xy) is 5ln|1+xy|+C