What is the integral of ((x+1))/(sqrt(x^2+2x+5)) ?

The integral of ((x+1))/(sqrt(x^2+2x+5)) is sqrt(x^2+2x+5)+C

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integral of ((x+1))/(sqrt(x^2+2x+5))

\int \frac{( x + 1 )}{x ^{2} + 2 x + 5} d x

x^{2} + 2 x + 5 + C

Solution steps

\int \frac{x + 1}{x ^{2} + 2 x + 5} d x

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

u = x^{2} + 2 x + 5

= \frac{1}{2} \cdot 2 (x^{2} + 2 x + 5)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (x^{2} + 2 x + 5)^{\frac{1}{2}} : x^{2} + 2 x + 5

= x^{2} + 2 x + 5

Add a constant to the solution

= x^{2} + 2 x + 5 + C

What is the integral of ((x+1))/(sqrt(x^2+2x+5)) ?
The integral of ((x+1))/(sqrt(x^2+2x+5)) is sqrt(x^2+2x+5)+C