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Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
Evaluate the following integrals:
$\int^\limits9_0\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}\sin\text{x},&0\leq\text{x}\leq\frac{\pi}{2}\\1,&\frac{\pi}{2}\leq\text{x}\leq3\\\text{e}^{\text{x}-3},&3\leq\text{x}\leq9\end{cases}$
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Answer

We have,
$\int^\limits9_0\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}\sin\text{x},&0\leq\text{x}\leq\frac{\pi}{2}\\1,&\frac{\pi}{2}\leq\text{x}\leq3\\\text{e}^{\text{x}-3},&3\leq\text{x}\leq9\end{cases}$
$\text{I}=\int^\limits9_0\text{f(x)}\text{dx}$
$\Rightarrow\text{I}=\int^\limits{\frac{\pi}{2}}_1\text{f(x)}\text{dx}+\int^\limits3_\frac{\pi}{2}\text{f(x)}\text{dx}+\int^\limits9_3\text{e}^{\text{x}-3}\text{ dx}$ [Additive property]
$\Rightarrow\text{I}=\int^\limits{\frac{\pi}{2}}_1\sin\text{x dx}+\int^\limits3_\frac{\pi}{2}\text{1 }\text{dx}+\int^\limits9_3\text{e}^{\text{x}-3}\text{ dx}$
$\Rightarrow\text{I}=\big[-\cos\text{x}\big]^{\frac{\pi}{2}}_0+\big[\text{x}\big]^3_\frac{\pi}{2}+\text{e}^6-\text{e}^0$
$\Rightarrow\text{I}=0+1+3-\frac{\pi}{2}+\text{e}^6-\text{e}^0$
$\Rightarrow\text{I}=3-\frac{\pi}{2}+\text{e}^6$

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