Question
Download Solution PDFThe value of \(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\) is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\)
Concept:
Apply formula :
\(\rm \int\frac{1}{\sqrt{1+t^2}}dt =\log\left(t+\sqrt{1+t^2}\right)+C\)
Calculation:
\(\int \frac{x^{\frac{3}{2}}}{\sqrt{1+x^5}} d x\)
\(\rm=\int \frac{x^{\frac{3}{2}}}{\sqrt{1+(x^\frac{5}{2})^2}} d x\)
Substitute \(\rm x^{\frac{5}{2}}=t\implies x^{\frac{3}{2}}dx=\frac{2}{5}dt\)
Then
\(\rm =\frac{2}{5}\int\frac{1}{\sqrt{1+t^2}}dt\)
\(\rm =\frac{2}{5}\log\left(t+\sqrt{1+t^2}\right)+C\)
Undo substitution
\(\rm =\frac{2}{5}\log\left(x^{\frac{5}{2}}+\sqrt{1+x^5}\right)+C\)
Hence option (3) is correct.
Last updated on Jun 19, 2025
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