What is the integral of 1/(usqrt(u^2+25)) ?

The integral of 1/(usqrt(u^2+25)) is 1/5 ln|tan((arctan(1/5 u))/2)|+C

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integral of 1/(usqrt(u^2+25))

\int \frac{1}{u u ^{2} + 25} d u

\frac{1}{5} ln tan (\frac{arctan ( \frac{1}{5} u )}{2}) + C

Solution steps

\int \frac{1}{u u ^{2} + 25} d u

Apply Trigonometric Substitution

: \int \frac{sec ( v )}{5 tan ( v )} d v

= \int \frac{sec ( v )}{5 tan ( v )} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{sec ( v )}{tan ( v )} d v

Apply u-substitution

: \int \frac{1}{w} d w

= \frac{1}{5} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= \frac{1}{5} ln ∣ w ∣

Substitute back

= \frac{1}{5} ln tan (\frac{arctan ( \frac{1}{5} u )}{2})

Add a constant to the solution

= \frac{1}{5} ln tan (\frac{arctan ( \frac{1}{5} u )}{2}) + C

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What is the integral of 1/(usqrt(u^2+25)) ?
The integral of 1/(usqrt(u^2+25)) is 1/5 ln|tan((arctan(1/5 u))/2)|+C