1.

Evaluate the following ​step by step

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Find the VALUE of the INTEGRAL cos^3x (LOWER limit = 0, upper limit = π/2)

Answer:

2/3

Step-by-step explanation:

\implies \sf{\int\limits^{\pi/2}_0 {cos^3x} \, dx }

\implies \sf{\int\limits^{\pi/2}_0 {(cos^2x)cosx} \, dx }

\implies \sf{\int\limits^{\pi/2}_0 {(1-sin^2x)cosx} \, dx }

\sf{Let\;sinx\: = t,}\\\sf{ differentiate\:w.r.t.\:x}\\\sf{cosx.dx = dt},\;\;\;\; therefore

\\\implies \sf{\int\limits^{\pi/2}_0 {1} \, dt - \int\limits^{\pi/2}_0 {t^2} \, dt}

\\\\\implies \sf{t|_0 ^{\pi/2} - \bigg[\dfrac{t^3}{3}\bigg]\bigg|_0 ^{\pi/2} }

\\\\\sf{\implies [sin(\pi/2) - sin(0)] - \dfrac{1}{3}[sin^3(\pi/2) - sin^3(0)]}

\\\\\sf{\implies (1-0) - (1/3) [1 - 0]}

\\\\\implies 1 - 1/3 \\\\\implies 2/3


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