Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Write a value of $\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}$
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Answer
$\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}$ Let $\text{t}=\sin\text{x}\rightarrow\text{dt}=\cos\text{x dx}$ $\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}=\int\text{e}^{\log\text{t}}\text{dt}=\text{I}$ $\text{e}^{\log\text{t}}\int1\text{dt}-\Big(\int\frac{\text{de}^{\log\text{t}}}{\text{dt}}\big(\int1\text{dt}\big)\text{dt}\Big)$ $=\text{e}^{\log\text{t}}\text{t}-\Big(\int\text{e}^{\log\text{t}}\frac{1}{\text{t}}\text{t dt}\Big)$ $=\text{e}^{\log\text{t}}\text{t}-\big(\int\text{e}^{\log\text{t}}\text{dt}\big)=\text{I}$ $\rightarrow\text{e}^{\log\text{t}}\text{t}-\text{I}=\text{I}\rightarrow2\text{I}=\text{e}^{\log\text{t}}+\text{C}$ $\text{I}=\frac{1}{2}\Big[\text{te}^{\log\text{t}}\Big]+\text{C}$ Substitute back $\text{t}=\sin\text{x}$ in above expression We get, $\text{I}=\frac{1}{2}\big[\sin{\text{x}}\text{e}^{\log\sin\text{x}}\big]+\text{C}$ $=\frac{\sin^2\text{x}}{2}+\text{C}$$[\because\log$ with base 10 term can be changed to in (natural log) term along with a constant$]$
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