Mathematics

If $$f(x) = \int_{0}^{x} t\sin t \ dt$$, write the value of $$f(x)$$


SOLUTION
$$\int _{ 0 }^{ x }{ t\sin t\quad dt } $$
Integrating by parts,
$$\Rightarrow $$$$\int { uv } =uv'-\int { v'\int { u } du } $$
$$u=t$$
$$v= sin t$$
$$\Rightarrow $$$$t\int {  sin t\ dt-\int { \sin t\ dt.\dfrac { d(t) }{ dt }  }  } $$
$$\Rightarrow $$$${ \left[ t\left( - \cos t \right)  \right]  }_{ 0 }^{ x }-{ \left[ \int { -\cos t dt }  \right]  }_{ 0 }^{ x }$$
$$\Rightarrow $$$${ \left[ -t \cos t \right]  }_{ 0 }^{ x }+{ \left[ \sin t \right]  }_{ 0 }^{ x }$$
$$\Rightarrow $$$$-x\cos x-(-0)+\sin x-\sin 0$$
$$\Rightarrow$$$$-x\cos x+\sin x$$

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Subjective Medium Published on 17th 09, 2020
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