What is the integral of (x+1)/(sqrt(x^2+2x+7)) ?

The integral of (x+1)/(sqrt(x^2+2x+7)) is sqrt(x^2+2x+7)+C

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integral of (x+1)/(sqrt(x^2+2x+7))

\int \frac{x + 1}{x ^{2} + 2 x + 7} d x

x^{2} + 2 x + 7 + C

Solution steps

\int \frac{x + 1}{x ^{2} + 2 x + 7} d x

Apply u-substitution

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

= \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

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

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

Substitute back

u = x^{2} + 2 x + 7

= \frac{1}{2} \cdot 2 (x^{2} + 2 x + 7)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (x^{2} + 2 x + 7)^{\frac{1}{2}} : x^{2} + 2 x + 7

= x^{2} + 2 x + 7

Add a constant to the solution

= x^{2} + 2 x + 7 + C

f (x) = 8 x^{5} - 8

e^{x}, - 1

\frac{d}{d x} (4 2 x)

f (x) = x^{2} (x + 1)

f (x) = \frac{2 x ^{2}}{x ^{2} - 2 x - 8}

What is the integral of (x+1)/(sqrt(x^2+2x+7)) ?
The integral of (x+1)/(sqrt(x^2+2x+7)) is sqrt(x^2+2x+7)+C