Question

If $f\left(x\right)={\int }_0^{x}t\sin tdt$, write the value of $f\left(x\right)$

Answer 1

${\int }_0^{x}t\sin tdt$

Integrating by parts,

$\Rightarrow$$\int uv=u{v}^{\prime }-\int v^{\prime }\int udu$

$u=t$

$v=sint$

$\Rightarrow$$t\int sintdt-\int \sin tdt.\frac{d\left(t\right)}{dt}$

$\Rightarrow$${\left[t(-\cos t)\right]}_0^x-{[\int -\cos tdt]}_0^x$

$\Rightarrow$${[-t\cos t]}_0^x+{\left[\sin t\right]}_0^x$

$\Rightarrow$$-x\cos x-(-0)+\sin x-\sin 0$

$\Rightarrow$$-x\cos x+\sin x$