# Find the indefinite integral of {eq}\int x^2 \arctan 3x dx{/eq}

## Question:

Find the indefinite integral of

{eq}\int x^2 \arctan 3x dx{/eq}

## Integration using By parts:

The integral with no limits is known as an indefinite integral. There is a constant of integration in the final answer of indefinite integral. If the function to be integrated is the product of two different types of functions, the method of by-parts is used. In this method, first, on the basis of a preference list, out of two functions, the first function ({eq}u {/eq}) and the second function, ({eq}v {/eq}) are selected. The preference list is 'ILATE'

where

{eq}I {/eq} is the inverse trigonometric function

{eq}L {/eq} is the logarithmic function

{eq}A {/eq} is the algebraic function

{eq}T {/eq} is the trigonometric function

{eq}E {/eq} is the exponential function.

The formula to determine the integral using by parts can be written as:

{eq}\int u v \ dx = u \int v \ dx - \int u' (\int v \ dx) \ dx {/eq}

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Given that: {eq}\displaystyle \int {{x^2}} \arctan 3xdx {/eq}.

{eq}\displaystyle \eqalign{ & \int {{x^2}} ta{n^{ - 1}}3xdx = \left(...