∫_π/2^3π/2[2sinx]dx Put x=π2+t⇒ dx=dt =∫_π/2^3π/2[2sin(π2+t)]dx =∫_π/2^3π/2[2cost]dt =∫_0^π/31 dt+∫_π/3^2π/30 dt+∫_π/2^2π/3( 1) dt+∫_2π ...

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Logarithmic Differentiation

∫π/23π/2[2sin...

Question

$\int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin x] d x$

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Solution

$\int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin x] d x$

Put $x = \frac{π}{2} + t \Rightarrow d x = d t$

$= \int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin (\frac{π}{2} + t)] d x$

$= \int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 cos t] d t$

$= \int_{0}^{\frac{π}{3}} 1 d t + \int_{\frac{π}{3}}^{\frac{2 π}{3}} 0 d t + \int_{\frac{π}{2}}^{\frac{2 π}{3}} (- 1) d t + \int_{\frac{2 π}{3}}^{π} (- 2) d t$

$= {[t]}_{0}^{\frac{π}{3}} + 0 - {[t]}_{\frac{π}{2}}^{\frac{2 π}{3}} - 2 {[t]}_{\frac{2 π}{3}}^{π}$

$= \frac{π}{3} - 0 + 0 - (\frac{2 π}{3} - \frac{π}{2}) - 2 [π - \frac{2 π}{3}]$

$= \frac{π}{3} - \frac{2 π}{3} + \frac{π}{2} - 2 π + \frac{4 π}{3}$

$= \frac{π - 2 π + 4 π}{3} + \frac{π}{2} - 2 π$

$= \frac{3 π}{3} + \frac{π}{2} - 2 π$

$= π + \frac{π}{2} - 2 π$

$= \frac{π}{2} - π$

$= \frac{π - 2 π}{2}$

$= \frac{- π}{2}$

$∴ \int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin x] d x = \frac{- π}{2}$

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