Question
Download Solution PDF\(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\) का मान है:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
\(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\)
संकल्पना:
सूत्र लागू करने पर:
\(\rm \int\frac{1}{\sqrt{1+t^2}}dt =\log\left(t+\sqrt{1+t^2}\right)+C\)
गणना:
\(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\)
\(\rm=\int \frac{x^{\frac{3}{2}}}{\sqrt{1+(x^\frac{5}{2})^2}} d x\)
\(\rm x^{\frac{5}{2}}=t\implies x^{\frac{3}{2}}dx=\frac{2}{5}dt\) रखने पर
तब
\(\rm =\frac{2}{5}\int\frac{1}{\sqrt{1+t^2}}dt\)
\(\rm =\frac{2}{5}\log\left(t+\sqrt{1+t^2}\right)+C\)
पुनः प्रतिस्थापन करने पर
\(\rm =\frac{2}{5}\log\left(x^{\frac{5}{2}}+\sqrt{1+x^5}\right)+C\)
अतः विकल्प (3) सही है।
Last updated on Jun 18, 2025
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