What is the integral of (12)/(1+9x^2) ?

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

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integral of (12)/(1+9x^2)

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

4 arctan (3 x) + C

Solution steps

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

Apply Integral Substitution

= 12 \cdot \int \frac{1}{3 ( 1 + u ^{2} )} 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 \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 12 \cdot \frac{1}{3} arctan (u)

Substitute back

u = 3 x

= 12 \cdot \frac{1}{3} arctan (3 x)

Simplify

12 \cdot \frac{1}{3} arctan (3 x) : 4 arctan (3 x)

= 4 arctan (3 x)

Add a constant to the solution

= 4 arctan (3 x) + C

\frac{d}{d x} (\frac{4 x ^{2} + 6 x + 2}{x})

ln (\frac{x}{x + a}), 1

\frac{d}{d x} (\frac{2 ^{x}}{2 ^{x} - 3 ^{x}})

\frac{d}{d x} (\frac{2 x + 3}{2 x})

\frac{d}{d x} (\frac{x}{sin ( x )})

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