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प्रश्न
\[\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx\] is equal to
पर्याय
xsin x + C
xsin x cos x + C
\[\frac{\left( x^{\sin x} \right)^2}{2} + C\]
none of these
MCQ
उत्तर
xsin x + C
\[\text{ Let I } = \int x^{\sin x} \left( \frac{\sin x}{x} + \cos x \cdot \log x \right)dx\]
\[\text{ Putting x}^{\sin x} = t\]
\[ \Rightarrow \ln \left( x \right)^{\sin x} = \ln t\]
\[ \Rightarrow \sin x \cdot \ln x = \ln t\]
\[ \Rightarrow \left( \sin x \times \frac{1}{x} + \cos x \ln x \right)dx = \frac{1}{t}dt\]
\[ \therefore I = \int t \cdot \frac{dt}{t}\]
\[ = t + C\]
\[ = x^{\sin x} + C .............\left( \because t = x^{\sin x} \right)\]
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