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Exercise

$\int x\tan^{-1}xdx$

Step-by-step Solution

1

We can solve the integral $\int x\arctan\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

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$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\arctan\left(x\right)}\\ \displaystyle{du=\frac{1}{1+x^2}dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=xdx}\\ \displaystyle{\int dv=\int xdx}\end{matrix}$
4

Solve the integral to find $v$

$v=\int xdx$
5

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{2}x^2$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{2}x^2\arctan\left(x\right)-\int\frac{x^2}{2\left(1+x^2\right)}dx$
7

Multiply the single term $2$ by each term of the polynomial $\left(1+x^2\right)$

$\frac{1}{2}x^2\arctan\left(x\right)-\int\frac{x^2}{2+2x^2}dx$
8

The integral $-\int\frac{x^2}{2+2x^2}dx$ results in: $-\frac{1}{2}x+\frac{1}{2}\arctan\left(x\right)$

$-\frac{1}{2}x+\frac{1}{2}\arctan\left(x\right)$
9

Gather the results of all integrals

$\frac{1}{2}x^2\arctan\left(x\right)+\frac{1}{2}\arctan\left(x\right)-\frac{1}{2}x$
10

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}x^2\arctan\left(x\right)+\frac{1}{2}\arctan\left(x\right)-\frac{1}{2}x+C_0$

Final answer to the exercise

$\frac{1}{2}x^2\arctan\left(x\right)+\frac{1}{2}\arctan\left(x\right)-\frac{1}{2}x+C_0$

Try other ways to solve this exercise

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  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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