Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- 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
- Load more...
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
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
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 $n=1$
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $2$ by each term of the polynomial $\left(1+x^2\right)$
The integral $-\int\frac{x^2}{2+2x^2}dx$ results in: $-\frac{1}{2}x+\frac{1}{2}\arctan\left(x\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
