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

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

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

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

x^{2} + 2 x + ln x^{2} + 2 x + x + 1 + C

Solution steps

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

Apply Integration By Parts

: x^{2} + 2 x - \int - \frac{1}{x ^{2} + 2 x} d x

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

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

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

Simplify

= x^{2} + 2 x + ln x^{2} + 2 x + x + 1

Add a constant to the solution

= x^{2} + 2 x + ln x^{2} + 2 x + x + 1 + C

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