Question

Assertion :$\int \frac{dx}{x^3\sqrt{1+x^4}}=-\frac{1}{2}\sqrt{1+\frac{1}{x^{{}^4}}}+C.$ Reason: For integration by parts, we have to follow ILATE rule.

• A. Assertion is True, Reason is True; Reason is a correct explanation for Assertion
• B. Assertion is True, Reason is True; Reason is NOT a correct explanation for Assertion
• C. Assertion is True, Reason is False
• D. Assertion is False, Reason is True

Answer 1

The correct option is B Assertion is True, Reason is True; Reason is NOT a correct explanation for Assertion

$I=\int \frac{dx}{x^3\sqrt{1+x^4}}$
$=\int \frac{dx}{x^5\sqrt{\frac{1}{x^{{}^4}}}+1}$
Put $\frac{1}{x^4}+1=t$
$dt=\frac{-4}{x^5}dx$
$\therefore I=-\frac{1}{4}\int \frac{dt}{\sqrt{t}}$
$=-\frac{1}{2}\sqrt{t}+C$
$=-\frac{1}{2}\sqrt{1+\frac{1}{x^4}}+C$
Hence, statement 1 is true.
Also, for integration by parts, we need to follow ILATE.
Thus, both the statements are true but statement 2 is not a correct explanation of statement 1.