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

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

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integral of x^2-1/((3x)^2)

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

\frac{x ^{3}}{3} + \frac{1}{9 x} + C

Solution steps

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

Simplify

x^{2} - \frac{1}{( 3 x ) ^{2}} : x^{2} - \frac{1}{9 x ^{2}}

= \int x^{2} - \frac{1}{9 x ^{2}} d x

Apply the Sum Rule:

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

= \int x^{2} d x - \int \frac{1}{9 x ^{2}} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{1}{9 x ^{2}} d x = - \frac{1}{9 x}

= \frac{x ^{3}}{3} - (- \frac{1}{9 x})

Simplify

= \frac{x ^{3}}{3} + \frac{1}{9 x}

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{1}{9 x} + C

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