What is the integral of 1/(sqrt(169+x^2)) ?

The integral of 1/(sqrt(169+x^2)) is ln(1/13 |x+sqrt(169+x^2)|)+C

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integral of 1/(sqrt(169+x^2))

\int \frac{1}{169 + x ^{2}} d x

ln (\frac{1}{13} x + 169 + x^{2}) + C

Solution steps

\int \frac{1}{169 + x ^{2}} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= \int sec (u) d u

Use the common integral:

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

= ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{1}{13} x)

= ln tan (arctan (\frac{1}{13} x)) + sec (arctan (\frac{1}{13} x))

Simplify

ln tan (arctan (\frac{1}{13} x)) + sec (arctan (\frac{1}{13} x)) : ln (\frac{1}{13} x + 169 + x^{2})

= ln (\frac{1}{13} x + 169 + x^{2})

Add a constant to the solution

= ln (\frac{1}{13} x + 169 + x^{2}) + C

\frac{\partial}{\partial x} (x^{2} ln (y) + 5 x y^{3})

x^{2} 8 x - 1

\frac{1 - 2 x}{4 + x}

\frac{d}{d x} (e^{x} cos (2 x) - 2 e^{x} sin (2 x))

\frac{\partial}{\partial y} (2 y)

What is the integral of 1/(sqrt(169+x^2)) ?
The integral of 1/(sqrt(169+x^2)) is ln(1/13 |x+sqrt(169+x^2)|)+C