Integral of xacosx dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = \operatorname{acos}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$.

   Then $\operatorname{du}{\left(x \right)} = - \frac{1}{\sqrt{1 - x^{2}}}$.

   To find $v{\left(x \right)}$:

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int x\, dx = \frac{x^{2}}{2}$$

   Now evaluate the sub-integral.

2. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \left(- \frac{x^{2}}{2 \sqrt{1 - x^{2}}}\right)\, dx = - \frac{\int \frac{x^{2}}{\sqrt{1 - x^{2}}}\, dx}{2}$$

   TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), restriction=(x > -1) & (x < 1), context=x**2/sqrt(1 - x**2), symbol=x)

   So, the result is: $- \frac{\begin{cases} - \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases}}{2}$

3. Now simplify:

   $$\begin{cases} \frac{2 x^{2} \operatorname{acos}{\left(x \right)} - x \sqrt{1 - x^{2}} + \operatorname{asin}{\left(x \right)}}{4} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$$

4. Add the constant of integration:

   $$\begin{cases} \frac{2 x^{2} \operatorname{acos}{\left(x \right)} - x \sqrt{1 - x^{2}} + \operatorname{asin}{\left(x \right)}}{4} & \text{for}\: x > -1 \wedge x < 1 \end{cases}+ \mathrm{constant}$$

The answer is:

$$\begin{cases} \frac{2 x^{2} \operatorname{acos}{\left(x \right)} - x \sqrt{1 - x^{2}} + \operatorname{asin}{\left(x \right)}}{4} & \text{for}\: x > -1 \wedge x < 1 \end{cases}+ \mathrm{constant}$$