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प्रश्न
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
विकल्प
2 cos \[\sqrt{x}\]
\[\sqrt{\frac{\cos x}{x}} + C\]
sin \[\sqrt{x} + C\]
2 sin \[\sqrt{x} + C\]
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उत्तर
2 sin \[\sqrt{x} + C\]
\[\text{Let }I = \int\frac{\cos \sqrt{x}}{\sqrt{x}}dx\]
\[\text{Putting }\sqrt{x} = t\]
\[ \Rightarrow \frac{1}{2\sqrt{x}}dx = dt\]
\[ \Rightarrow \frac{dx}{\sqrt{x}} = 2dt\]
\[ \therefore I = 2\int\cos t \cdot dt\]
\[ = 2 \sin t + C\]
\[ = 2 \sin \sqrt{x} + C ..................\left(\because t = \sqrt{x} \right)\]
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