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Standard Formulae - 2

The integral ...

Question

The integral $\int_{\frac{π}{4}}^{\frac{5 π}{4}}$ $(| c o s t | s i n t + | s i n t | c o s t) d t$ has the value equal to ‘k’ then [k] is ([.] denotes the greatest integer function).___

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Solution

$I = \int_{\frac{π}{4}}^{π} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (- s i n t c o s t) +      (z e r o) (s i n t c o s t) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ d t + \int_{π}^{\frac{5 π}{4}} - 2 s i n t c o s t d t$
$= \int_{\frac{π}{4}}^{π} s i n 2 t d t - \int_{π}^{\frac{5 π}{4}} - s i n 2 t d t = 0$ these two integrals cancels $\Rightarrow$ k = 0

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