What is the integral of 1/(sqrt(x^2+4x+53)) ?

The integral of 1/(sqrt(x^2+4x+53)) is ln(1/7 |x+2+sqrt(x^2+4x+53)|)+C

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integral of 1/(sqrt(x^2+4x+53))

\int \frac{1}{x ^{2} + 4 x + 53} d x

ln (\frac{1}{7} x + 2 + x^{2} + 4 x + 53) + C

Solution steps

\int \frac{1}{x ^{2} + 4 x + 53} d x

Complete the square

x^{2} + 4 x + 53 : (x + 2)^{2} + 49

= \int \frac{1}{( x + 2 ) ^{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 + 2))) + sec (arctan (\frac{1}{7} (x + 2)))

Simplify

ln tan (arctan (\frac{1}{7} (x + 2))) + sec (arctan (\frac{1}{7} (x + 2))) : ln (\frac{1}{7} x + 2 + x^{2} + 4 x + 53)

= ln (\frac{1}{7} x + 2 + x^{2} + 4 x + 53)

Add a constant to the solution

= ln (\frac{1}{7} x + 2 + x^{2} + 4 x + 53) + C

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What is the integral of 1/(sqrt(x^2+4x+53)) ?
The integral of 1/(sqrt(x^2+4x+53)) is ln(1/7 |x+2+sqrt(x^2+4x+53)|)+C