Solutions - 0010-02

(1) Select $u$ and $v^{\prime }$ considering LIATE:

$$\begin{array}{cc}u=x^2& v^{\prime }=\sin x\\ u^{\prime }=2x& v=-\cos x\end{array}\begin{array}{c}⟶\int u\cdot v^{\prime }\text{original}\\ ⟶\int u^{\prime }\cdot v\text{final}\end{array}$$

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(2) Apply IBP formula $uv-\int u^{\prime }\cdot v$:

$$\begin{array}{cccc}& uv-\int u^{\prime }\cdot v& \gg \gg x^2(-\cos x)-\int 2x(-\cos x)dx& \\ & & & \\ & & \gg \gg -x^2\cos x+2\int x\cos xdx& \text{(Exp. A)}\end{array}$$

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(3) Select another $u=x$ and $v^{\prime }=\cos x$ and do IBP again:

$$\begin{array}{cc}\int x\cos x& \gg \gg x\sin x-\int \sin xdx\\ & \\ & \gg \gg x\sin x+\cos x+C\end{array}$$

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(4) Put all together in (A):

$$\begin{array}{ccc}& \gg \gg -x^2\cos x+2(x\sin x+\cos x+C)& \\ & & \\ & \gg \gg -x^2\cos x+2x\sin x+2\cos x+C& \text{(Note B)}\end{array}$$

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Note B: We can change notation $2C$ to $C$ because the value of $C$ is arbitrary.