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Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
Evaluate the following integrals:$\int^\limits4_1\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}7\text{x}+3,&\text{if }\ 1\leq\text{x}\leq3\\8\text{x},&\text{if }\ 3\leq\text{x}\leq9\end{cases}$
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Answer

We have,$\int^\limits4_1\text{f(x)}\text{dx},$ Where $\text{f(x)}=\begin{cases}7\text{x}+3,&\text{if }\ 1\leq\text{x}\leq3\\8\text{x},&\text{if }\ 3\leq\text{x}\leq9\end{cases}$
$\text{I}=\int^\limits4_1\text{f(x)}\text{dx}$
$\Rightarrow\text{I}=\int^\limits{3}_1\text{f(x)}\text{dx}+\int^\limits4_3\text{f(x)}\text{dx}$ [Additive property]
$\Rightarrow\text{I}=\int^\limits{3}_1(7\text{x}+3)\text{dx}+\int^\limits{4}_38\text{x dx}$
$\Rightarrow\text{I}=\Big[\frac{7\text{x}^2}{2}+3\text{x}\big]^3_1+\big[4\text{x}^2\big]^4_2$
$\Rightarrow\text{I}=\frac{63}{2}+9-\frac{7}{2}-3+64-36$
$\Rightarrow\text{I}=\frac{56}{2}+34$
$\Rightarrow\text{I}=62$

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