Question

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

We start solving the problem by considering the indefinite integral given in the Assertion. We then make the necessary arrangements in the integral and assume $\dfrac{1}{{{x}^{4}}}+1=t$ and find the $dx$ in terms of $dt$. We then make use of the result $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C$, $n\ne -1$ and make the necessary calculations to check the result of assertion. We then check whether the given Reason can exactly explain the given Assertion to get the required answer.

Complete step by step answer:
According to the problem, we are given an Assertion as $\int{\dfrac{dx}{{{x}^{3}}\sqrt{1+{{x}^{4}}}}}=\dfrac{-1}{2}\sqrt{1+\dfrac{1}{{{x}^{4}}}}+C$ and Reason as: For integration by parts, we have to follow ILATE rule. We then check whether they are true and also check the relation between them.
Let us first solve the statement given Assertion.
Let us assume $I=\int{\dfrac{dx}{{{x}^{3}}\sqrt{1+{{x}^{4}}}}}$.
$\Rightarrow I=\int{\dfrac{dx}{{{x}^{3}}\sqrt{{{x}^{4}}\left( \dfrac{1}{{{x}^{4}}}+1 \right)}}}$.
$\Rightarrow I=\int{\dfrac{dx}{{{x}^{3}}\times {{x}^{2}}\sqrt{\left( \dfrac{1}{{{x}^{4}}}+1 \right)}}}$.
$\Rightarrow I=\int{\dfrac{dx}{{{x}^{5}}\sqrt{\left( \dfrac{1}{{{x}^{4}}}+1 \right)}}}$ ---(1).
Let us assume $\dfrac{1}{{{x}^{4}}}+1=t$ ---(2).
Let us apply a differential on both sides of equation (2).
$\Rightarrow d\left( \dfrac{1}{{{x}^{4}}}+1 \right)=d\left( t \right)$.
$\Rightarrow d\left( \dfrac{1}{{{x}^{4}}} \right)+d\left( 1 \right)=dt$.
$\Rightarrow \left( \dfrac{-4}{{{x}^{5}}} \right)dx=dt$.
$\Rightarrow \dfrac{dx}{{{x}^{5}}}=\dfrac{-dt}{4}$ ---(3).
Let us substitute equations (2) and equation (3) in equation (1).
$\Rightarrow I=\int{\dfrac{-dt}{4\sqrt{t}}}$.
$\Rightarrow I=\dfrac{-1}{4}\int{{{t}^{\dfrac{-1}{2}}}dt}$.
We know that $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C$, $n\ne -1$.
$\Rightarrow I=\dfrac{-1}{4}\times \left( \dfrac{{{t}^{\dfrac{-1}{2}+1}}}{\dfrac{-1}{2}+1} \right)+C$.
$\Rightarrow I=\dfrac{-1}{4}\times \left( \dfrac{{{t}^{\dfrac{1}{2}}}}{\dfrac{1}{2}} \right)+C$.
$\Rightarrow I=\dfrac{-1}{2}\times \sqrt{t}+C$.
From equation (2), we get
$\Rightarrow I=\dfrac{-1}{2}\sqrt{1+\dfrac{1}{{{x}^{4}}}}+C$.
So, we have found Assertion is True.
We know that we follow ILATE rules while performing integration by parts. So, we have found Reason is also true.
We can see that we have not used integration by parts to solve the indefinite integral given in Assertion. So, Reason is not a correct explanation of Assertion.

So, the correct answer is “Option b”.

Note: Whenever we get this type of problems, we first check whether the given statements are true or false and then check if there is a relation between them. We should not forget to add constants of integration while solving problems related to indefinite integrals. We should not forget to substitute the function related to ‘t’ after finding the result in ‘t’. Similarly, we can expect problems to find the value of the indefinite integral $\int{x{{\sin }^{-1}}xdx}$ as Assertion.