Evaluate the indefinite integral.

{eq}\int x \ {arctan }(4x)dx {/eq}

Question:

Integration by Parts:

Integration by parts method is used to compute the antiderivative of the product of two functions such as,

$$\displaystyle \int {a\left( x \right) \times b\left( x \right)} \;dx $$

Here, {eq}a(x) {/eq} and {eq}b(x) {/eq} are both different and differentiable functions.

In this method, we apply the following integration formula to find the solution of that type of integrals.

$$\displaystyle \int {a\left( x \right) \times b\left( x \right)} \;dx = a\left( x \right)\int {b\left( x \right)} \;dx - \int {\left[ {\dfrac{d}{{dx}}a\left( x \right)\int {b\left( x \right)} \;dx} \right]\;dx} $$

Here, {eq}a(x) {/eq} and {eq}b(x) {/eq} are the first and second functions respectively.

It is also known as the partial integration method. In this method, we also apply the ILATE rule find the first and second functions. The first function has higher priority rather than second function. The following expression helps us to choose the first and second functions.

$${\rm{Inverse}}\;{\rm{trigonometric}}\;{\rm{function}}\left( {{\rm{Top}}\;{\rm{priority}}} \right) > {\rm{Logarithmic}}\;{\rm{function}} > {\rm{Algebraic}}\;{\rm{function}} > {\rm{Trigonometric}}\;{\rm{function}} > {\rm{Exponential}}\;{\rm{function}}\left( {{\rm{Lowest}}\;{\rm{priority }}} \right) $$

Answer and Explanation: