What is the integral of 5/(sqrt(3s^2+1)) ?

The integral of 5/(sqrt(3s^2+1)) is 5/(sqrt(3))ln|sqrt(3)s+sqrt(1+3s^2)|+C

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integral of 5/(sqrt(3s^2+1))

\int \frac{5}{3 s ^{2} + 1} d s

\frac{5}{3} ln 3 s + 1 + 3 s^{2} + C

Solution steps

\int \frac{5}{3 s ^{2} + 1} d s

Take the constant out:

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

= 5 \cdot \int \frac{1}{3 s ^{2} + 1} d s

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{3} d u

= 5 \cdot \int \frac{sec ( u )}{3} d u

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \int sec (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{3} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (3 s)

= 5 \cdot \frac{1}{3} ln tan (arctan (3 s)) + sec (arctan (3 s))

Simplify

5 \cdot \frac{1}{3} ln tan (arctan (3 s)) + sec (arctan (3 s)) : \frac{5}{3} ln 3 s + 1 + 3 s^{2}

= \frac{5}{3} ln 3 s + 1 + 3 s^{2}

Add a constant to the solution

= \frac{5}{3} ln 3 s + 1 + 3 s^{2} + C

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What is the integral of 5/(sqrt(3s^2+1)) ?
The integral of 5/(sqrt(3s^2+1)) is 5/(sqrt(3))ln|sqrt(3)s+sqrt(1+3s^2)|+C