Question
Download Solution PDF\(\rm \int_{1}^{e}\frac{dx}{x\sqrt{2+\ln x}}\) का मूल्यांकन कीजिए।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
\(\rm \dfrac{d(\ln x)}{dx} = \dfrac{1}{x}\)
गणना:
माना कि I = \(\rm \int_{1}^{e}\frac{dx}{x\sqrt{2+\ln x}}\) है।
माना कि (2 + ln x) = t2 है।
x के संबंध में अवकलन करने पर, हमें निम्न प्राप्त होता है
⇒ (0 + \(\rm \frac 1 x\))dx = 2tdt
⇒ \(\rm \frac 1 x\)dx = 2tdt
|
x |
1 |
e |
|
t |
\(\sqrt 2\) |
\(\sqrt 3\) |
अब,
\(\rm I=\rm \int_{\sqrt 2}^{\sqrt 3}\frac{2tdt}{\sqrt{t^2}}\\=2\int_{\sqrt 2}^{\sqrt 3}\frac{tdt}{t}\\=2\int_{\sqrt 2}^{\sqrt 3}dt\\=2[t]_{\sqrt 2}^{\sqrt 3}\\=2(\sqrt 3- \sqrt 2)\)
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