What is the integral of (12)/(sqrt(4+9x^2)) ?

The integral of (12)/(sqrt(4+9x^2)) is 4ln(1/2 |3x+sqrt(9x^2+4)|)+C

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integral of (12)/(sqrt(4+9x^2))

\int \frac{12}{4 + 9 x ^{2}} d x

4 ln (\frac{1}{2} 3 x + 9 x^{2} + 4) + C

Solution steps

\int \frac{12}{4 + 9 x ^{2}} d x

Take the constant out:

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

= 12 \cdot \int \frac{1}{4 + 9 x ^{2}} d x

Apply Trigonometric Substitution

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

= 12 \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

= 12 \cdot \frac{1}{3} \cdot \int sec (u) d u

Use the common integral:

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

= 12 \cdot \frac{1}{3} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{3}{2} x)

= 12 \cdot \frac{1}{3} ln tan (arctan (\frac{3}{2} x)) + sec (arctan (\frac{3}{2} x))

Simplify

12 \cdot \frac{1}{3} ln tan (arctan (\frac{3}{2} x)) + sec (arctan (\frac{3}{2} x)) : 4 ln (\frac{1}{2} 3 x + 9 x^{2} + 4)

= 4 ln (\frac{1}{2} 3 x + 9 x^{2} + 4)

Add a constant to the solution

= 4 ln (\frac{1}{2} 3 x + 9 x^{2} + 4) + C

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What is the integral of (12)/(sqrt(4+9x^2)) ?
The integral of (12)/(sqrt(4+9x^2)) is 4ln(1/2 |3x+sqrt(9x^2+4)|)+C