What is the integral of sqrt(1+cot^2(5x)) ?

The integral of sqrt(1+cot^2(5x)) is 1/5 ln|tan((5x)/2)|+C

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integral of sqrt(1+cot^2(5x))

\int 1 + cot^{2} (5 x) d x

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

Solution steps

\int 1 + cot^{2} (5 x) d x

Apply u-substitution

: \int \frac{1}{5} csc (u) d u

= \int \frac{1}{5} csc (u) d u

Take the constant out:

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

= \frac{1}{5} \cdot \int csc (u) d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{5} ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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

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What is the integral of sqrt(1+cot^2(5x)) ?
The integral of sqrt(1+cot^2(5x)) is 1/5 ln|tan((5x)/2)|+C