Evaluate ∫ 5 9 cos − 1 ⁡ ( t ) d t.

Question

Evaluate   ∫      5    9        cos          −      1        ⁡  (  t  )  d  t.

Jerry Kidd · Accepted Answer

Since&amp;nbsp;59&amp;nbsp;is constant with respect to&amp;nbsp;t, move&amp;nbsp;59&amp;nbsp;out of the integral.59∫cos-1(t)dtIntegrate by parts using the formula&amp;nbsp;∫udv=uv-∫vdu, where&amp;nbsp;u=cos-1(t)&amp;nbsp;and&amp;nbsp;dv=1.59(cos-1(t)t-∫t(-11-t2)dt)Combine&amp;nbsp;t&amp;nbsp;and&amp;nbsp;11-t2.59(cos-1(t)t-∫-t1-t2dt)Since&amp;nbsp;-1&amp;nbsp;is constant with respect to&amp;nbsp;t, move&amp;nbsp;-1&amp;nbsp;out of the integral.59(cos-1(t)t--∫t1-t2dt)Simplify.59(cos-1(t)t+∫t1-t2dt)Let&amp;nbsp;u=1-t2. Then&amp;nbsp;du=-2tdt, so&amp;nbsp;-12du=tdt. Rewrite using&amp;nbsp;u&amp;nbsp;and&amp;nbsp;du.59(cos-1(t)t+∫1u⋅1-2du)Simplify.59(cos-1(t)t+∫-12udu)Since&amp;nbsp;-1&amp;nbsp;is constant with respect to&amp;nbsp;u, move&amp;nbsp;-1&amp;nbsp;out of the integral.59(cos-1(t)t-∫12udu)Since&amp;nbsp;12&amp;nbsp;is constant with respect to&amp;nbsp;u, move&amp;nbsp;12&amp;nbsp;out of the integral.59(cos-1(t)t-(12∫1udu))Apply basic rules of exponents.59(cos-1(t)t-12∫u-12du)By the Power Rule, the integral of&amp;nbsp;u-12&amp;nbsp;with respect to&amp;nbsp;u&amp;nbsp;is&amp;nbsp;2u12.59(cos-1(t)t-12(2u12+C))Rewrite&amp;nbsp;59(cos-1(t)t-12(2u12+C))&amp;nbsp;as&amp;nbsp;59(cos-1(t)t-u12)+C.59(cos-1(t)t-u12)+CReplace all occurrences of&amp;nbsp;u&amp;nbsp;with&amp;nbsp;1-t2.59(cos-1(t)t-(1-t2)12)+CSimplify.5(tcos-1(t)-(1-t2)12)9+CReorder terms.59(tcos-1(t)-(1-t2)12)+C

Evaluate 5 9 cos − 1 </mrow> ⁡<!-- ⁡

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