What is the integral of (3x^2)/(2(9x^2+1)) ?

The integral of (3x^2)/(2(9x^2+1)) is 1/18 (-arctan(3x)+3x)+C

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integral of (3x^2)/(2(9x^2+1))

\int \frac{3 x ^{2}}{2 ( 9 x ^{2} + 1 )} d x

\frac{1}{18} (- arctan (3 x) + 3 x) + C

Solution steps

\int \frac{3 x ^{2}}{2 ( 9 x ^{2} + 1 )} d x

Take the constant out:

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

= \frac{3}{2} \cdot \int \frac{x ^{2}}{9 x ^{2} + 1} d x

Apply Integral Substitution

= \frac{3}{2} \cdot \int \frac{u ^{2}}{27 ( u ^{2} + 1 )} d u

Take the constant out:

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

= \frac{3}{2} \cdot \frac{1}{27} \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= \frac{3}{2} \cdot \frac{1}{27} \cdot \int - \frac{1}{u ^{2} + 1} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{2} \cdot \frac{1}{27} (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

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

\int 1 d u = u

= \frac{3}{2} \cdot \frac{1}{27} (- arctan (u) + u)

Substitute back

u = 3 x

= \frac{3}{2} \cdot \frac{1}{27} (- arctan (3 x) + 3 x)

Simplify

\frac{3}{2} \cdot \frac{1}{27} (- arctan (3 x) + 3 x) : \frac{1}{18} (- arctan (3 x) + 3 x)

= \frac{1}{18} (- arctan (3 x) + 3 x)

Add a constant to the solution

= \frac{1}{18} (- arctan (3 x) + 3 x) + C

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What is the integral of (3x^2)/(2(9x^2+1)) ?
The integral of (3x^2)/(2(9x^2+1)) is 1/18 (-arctan(3x)+3x)+C