Final answer to the problem
$0.4388246$
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Step-by-step Solution
Specify the solving method
1
We can solve the integral $\int\arctan\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2
First, identify $u$ and calculate $du$
$\begin{matrix}\displaystyle{u=\arctan\left(x\right)}\\ \displaystyle{du=\frac{1}{1+x^2}dx}\end{matrix}$
3
Now, identify $dv$ and calculate $v$
$\begin{matrix}\displaystyle{dv=1dx}\\ \displaystyle{\int dv=\int 1dx}\end{matrix}$
4
Solve the integral
$v=\int1dx$
5
The integral of a constant is equal to the constant times the integral's variable
$x$
6
Now replace the values of $u$, $du$ and $v$ in the last formula
$\left[x\arctan\left(x\right)\right]_{0}^{1}-\int_{0}^{1}\frac{x}{1+x^2}dx$
7
The integral $-\int_{0}^{1}\frac{x}{1+x^2}dx$ results in: $-\frac{96}{277}$
$-\frac{96}{277}$
8
Gather the results of all integrals
$\left[x\arctan\left(x\right)\right]_{0}^{1}-\frac{96}{277}$
9
Evaluate the definite integral
$1\arctan\left(1\right)- 0\arctan\left(0\right)-\frac{96}{277}$
10
Simplify the expression inside the integral
$0.4388246$
Final answer to the problem
$0.4388246$