Question

Assertion :$\int \frac{x^{5/2}}{\sqrt{1+x^7}}dx$ is equal to $\frac{2}{7}\log |x^{7/2}+\sqrt{1+x^7}|+C$ Reason: $\int \frac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{1+x^2}|+C$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\int \frac{x^{5/2}}{\sqrt{1+x^7}}dx$
$=\int \frac{x^{5/2}}{\sqrt{1+{\left(x^{7/2}\right)}^2}}dx$
Putting $x^{7/2}=t$
$\Rightarrow \frac{7}{2}{x}^{5/2}dx=dt$
$\therefore I=\frac{2}{7}\int \frac{dt}{\sqrt{1+t^2}}$
$=\frac{2}{7}\log |t+\sqrt{1+t^2}|+C$ ($\because \int \frac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{1+x^2}|+C$)
$=\frac{2}{7}\log |x^{\frac{7}{2}}+\sqrt{1+x^7}|+C$
$\therefore$ Reason (R) is the correct explanation for Assertion (A).