use integration by parts to evaluate the indefinite integral ∫ 4x sin(x) dx.

Explanation:

Step1: Recall integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = 4x$ and $\mathrm{d}v=\sin(x)\mathrm{d}x$.

Step2: Find $\mathrm{d}u$ and $v$

Differentiate $u = 4x$ with respect to $x$ to get $\mathrm{d}u = 4\mathrm{d}x$. Integrate $\mathrm{d}v=\sin(x)\mathrm{d}x$ with respect to $x$ to get $v=-\cos(x)$.

Step3: Apply the integration - by - parts formula

$\int 4x\sin(x)\mathrm{d}x=4x(-\cos(x))-\int(-\cos(x))\times4\mathrm{d}x$.

Step4: Simplify the right - hand side

$=-4x\cos(x)+4\int\cos(x)\mathrm{d}x$.

Step5: Integrate $\cos(x)$

Since $\int\cos(x)\mathrm{d}x=\sin(x)+C$, then $-4x\cos(x)+4\int\cos(x)\mathrm{d}x=-4x\cos(x) + 4\sin(x)+C$.

Answer:

$-4x\cos(x)+4\sin(x)+C$