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

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

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

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

\frac{1}{2} x^{2} + 2 x + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{2} + 2 x

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} \cdot 2 (x^{2} + 2 x)^{\frac{1}{2}} : \frac{1}{2} x^{2} + 2 x

= \frac{1}{2} x^{2} + 2 x

Add a constant to the solution

= \frac{1}{2} x^{2} + 2 x + C

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