What is the integral of 1/(sqrt(x^2+2x+50)) ?

The integral of 1/(sqrt(x^2+2x+50)) is ln(1/7 |x+1+sqrt(x^2+2x+50)|)+C

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integral of 1/(sqrt(x^2+2x+50))

\int \frac{1}{x ^{2} + 2 x + 50} d x

ln (\frac{1}{7} x + 1 + x^{2} + 2 x + 50) + C

Solution steps

\int \frac{1}{x ^{2} + 2 x + 50} d x

Complete the square

x^{2} + 2 x + 50 : (x + 1)^{2} + 49

= \int \frac{1}{( x + 1 ) ^{2} + 49} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 49} d u

= \int \frac{1}{u ^{2} + 49} d u

Apply Trigonometric Substitution

: \int sec (v) d v

= \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

Substitute back

= ln tan (arctan (\frac{1}{7} (x + 1))) + sec (arctan (\frac{1}{7} (x + 1)))

Simplify

ln tan (arctan (\frac{1}{7} (x + 1))) + sec (arctan (\frac{1}{7} (x + 1))) : ln (\frac{1}{7} x + 1 + x^{2} + 2 x + 50)

= ln (\frac{1}{7} x + 1 + x^{2} + 2 x + 50)

Add a constant to the solution

= ln (\frac{1}{7} x + 1 + x^{2} + 2 x + 50) + C

x \to 1 + lim (\frac{2 + x}{1 - x})

x \to 3 lim (2 - \frac{1}{x})

\frac{d}{d x} (sin (\frac{1}{5} x))

x \to 2 lim (2 x + 3)

tan (2 (x + 1))

What is the integral of 1/(sqrt(x^2+2x+50)) ?
The integral of 1/(sqrt(x^2+2x+50)) is ln(1/7 |x+1+sqrt(x^2+2x+50)|)+C