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Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
If $[\cdot]$ and $\{\cdot\}$ denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:$\int\limits^{\frac{\pi}{4}}_0\sin\{\text{x}\}\text{dx}$
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Answer

We have,$\text{I}=\int\limits^{\frac{\pi}{4}}_0\sin\{\text{x}\}\text{dx}$
We know that,$\{\text{x}\}=\text{x},\text{ when }0<\text{x}<\frac{\pi}{4}$ $\big(\text{As }\pi=3.14\Rightarrow\frac{\pi}{4}=0.784<1\big)$
$\therefore\ \text{I}=\int\limits^{\frac{\pi}{4}}_0\sin\{\text{x}\}\text{dx}$
$=\big[-\cos\text{x}\big]^{\frac{\pi}{4}}_{0}$
$=-\Big(\cos\frac{\pi}{4}-\cos\frac{\pi}{4}\Big)$
$=\cos0-\cos\frac{\pi}{4}$
$=1-\frac{1}{\sqrt{2}}$
$=\frac{\sqrt{2}-1}{\sqrt{2}}$

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