What is the integral of 1/(sqrt(441+x^2)) ?

The integral of 1/(sqrt(441+x^2)) is ln(1/21 |x+sqrt(441+x^2)|)+C

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integral of 1/(sqrt(441+x^2))

\int \frac{1}{441 + x ^{2}} d x

ln (\frac{1}{21} x + 441 + x^{2}) + C

Solution steps

\int \frac{1}{441 + 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}{21} x)

= ln tan (arctan (\frac{1}{21} x)) + sec (arctan (\frac{1}{21} x))

Simplify

ln tan (arctan (\frac{1}{21} x)) + sec (arctan (\frac{1}{21} x)) : ln (\frac{1}{21} x + 441 + x^{2})

= ln (\frac{1}{21} x + 441 + x^{2})

Add a constant to the solution

= ln (\frac{1}{21} x + 441 + x^{2}) + C

\int - \frac{x ^{3}}{4} d x

\int (x^{2} + x) e^{x} d x

x \to 0 lim ((1 - 9 x)^{\frac{1}{x}})

\frac{d}{d x} (4 9 x^{2} + 5)

x^{2} + y^{2}

What is the integral of 1/(sqrt(441+x^2)) ?
The integral of 1/(sqrt(441+x^2)) is ln(1/21 |x+sqrt(441+x^2)|)+C