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 the 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

We can see that the given integral in assertion is indefinite and we make the necessary arguments in the integral and assume $1 + \dfrac{1}{{{x^4}}} = t$ and find dx in terms of dt and then use the result $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C}$, $n \ne - 1$ and after solving compare with R.H.S. of the assertion. We then check whether the given reason can explain the given assertion to get the required answer.

Complete step by step solution:
We first try to solve the given assertion in the question $\int {\dfrac{{dx}}{{{x^3}\sqrt {1 + {x^4}} }} = - \dfrac{1}{2}\sqrt {1 + \dfrac{1}{{{x^4}}}} + C}$ , and reason: For integration by parts, we have to follow the ILATE rule. Then we check the result and compare whether the given reason is correct or not.
Let us first try to solve the given assertion.
Let us assume $I = \int {\dfrac{{dx}}{{{x^3}\sqrt {1 + {x^4}} }}}$
Taking ${x^4}$ common from the integration I
$\Rightarrow I = \int {\dfrac{{dx}}{{{x^3}\sqrt {{x^4}(1 + \dfrac{1}{{{x^4}}})} }}}$
$\Rightarrow I = \int {\dfrac{{dx}}{{{x^3} \times {x^2}\sqrt {1 + \dfrac{1}{{{x^4}}}} }}}$
$\Rightarrow I = \int {\dfrac{{dx}}{{{x^5}\sqrt {1 + \dfrac{1}{{{x^4}}}} }}}$………….. (1)
Now let us assume that $1 + \dfrac{1}{{{x^4}}} = t$…………………. (2)
$\therefore$On differentiating the equation on both sides, We get:
$\Rightarrow d\left( {1 + \dfrac{1}{{{x^4}}}} \right) = d\left( t \right)$
$\Rightarrow d\left( {\dfrac{1}{{{x^4}}}} \right) + d\left( 1 \right) = d\left( t \right)$
As we know that differentiation of any constant is zero and the differentiation of $d\left( {{x^n}} \right) = n{x^{n - 1}}$
So, We get:
$\Rightarrow \left( {\dfrac{{ - 4}}{{{x^5}}}} \right)dx + 0 = dt$
$\Rightarrow \dfrac{{dx}}{{{x^5}}} = - \dfrac{{dt}}{4}$…………………. (3)
Now, substitute equation (2) and equation (3) in equation (1), we get:
$\Rightarrow I = - \int {\dfrac{{dt}}{{4\sqrt t }}}$
$\Rightarrow I = - \dfrac{1}{4}\int {{t^{\dfrac{{ - 1}}{2}}}dt}$
Now use the same result that $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C}$, $n \ne - 1$ and we get:
$\Rightarrow I = - \dfrac{1}{4}\left( {\dfrac{{{t^{\dfrac{{ - 1}}{2} + 1}}}}{{\dfrac{{ - 1}}{2} + 1}}} \right) + C$
$\Rightarrow I = - \dfrac{1}{4}\left( {\dfrac{{{t^{\dfrac{1}{2}}}}}{{\dfrac{1}{2}}}} \right) + C$
$\Rightarrow I = - \dfrac{1}{2} \times \sqrt t + C$
After putting the value of equation (2) $1 + \dfrac{1}{{{x^4}}} = t$ , we get:
$\Rightarrow I = - \dfrac{1}{2} \times \sqrt {1 + \dfrac{1}{{{x^4}}}} + C$
So, we can see that the result we get is the same as given in the assertion of the question>
So, the given assertion is true.
After solving this equation we see that there is no use of the ILATE method to solve this indefinite integration.
But when we perform Integration by parts, the ILATE rule is used. So, the reason is also true but in this question, we do not use integration by parts using the ILATE method. So the reason is not the correct explanation for the given assertion.

$\therefore$ Option (B) is correct.

Note:
In these types of problems, we try to solve the given assertion that it is true or false. After solving Assertion we try to find the relation between assertion and reason.
Also, do not forget to add the constant in the integral.