The value of ∫014x3d2dx21-x25dx is ____________

JEE Advanced · Mathematics · 27. Definite Integration

The value of $\int_{0}^{1} 4 x^{3} \{\frac{d^{2}}{d x^{2}} {(1 - x^{2})}^{5}\} d x$ is ____________

Verified Solution

Answer & Solution

Correct Answer

(A) 2

Step-by-step Solution

Detailed explanation

\int_{0}^{1} \underset{⏟}{4 x^{3}} \underset{⏟}{\frac{d^{2}}{d x^{2}} {(1 - x^{2})}^{5}} d x

= {[4 x^{3} \frac{d}{d x} {(1 - x^{2})}^{5}]}_{0}^{1} - \int_{0}^{1} 12 x^{2} \frac{d}{d x} {(1 - x^{2})}^{5} d x

= {[4 x^{3} \times 5 {(1 - x^{2})}^{4} (- 2 x)]}_{0}^{1} - 12 [{[x^{2} {(1 - x^{2})}^{5}]}_{0}^{1} - \int_{0}^{1} 2 x {(1 - x^{2})}^{5} d x]

= 0 - 0 - 12 [0 - 0] + 12 \int_{0}^{1} 2 x {(1 - x^{2})}^{5} d x

= 12 \times {[- \frac{{(1 - x^{2})}^{6}}{6}]}_{0}^{1}

= 12 [0 + \frac{1}{6}] = 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

Match the following

Column I	Column II
(A) $CH _3- CHBr - CD _3$ on treatment with alc. KOH gives $CH _2= CH - CD _3$ as a major product.	(P) E1 reaction
(B) $Ph - CHBr - CH _3$ reacts faster than $Ph - CHBr - CD _3$	(Q) E2 reaction
(C) $Ph - CH _2- CH _2 Br$ on treatment with $C _2 H _5 OD / C _2 H _5 O ^{-}$gives $Ph - CD = CH _2$ as the major product	(R) El cb reaction
(D) $PhCH _2 CH _2 Br$ and $PhCH _2 CH _2 Br$ react with same rate	(S) First order reaction

JEE Advanced 2006 Medium

Explore more questions on app

Column I	Column II
(A) \(CH _3- CHBr - CD _3\) on treatment with alc. KOH gives \(CH _2= CH - CD _3\) as a major product.	(P) E1 reaction
(B) \(Ph - CHBr - CH _3\) reacts faster than \(Ph - CHBr - CD _3\)	(Q) E2 reaction
(C) \(Ph - CH _2- CH _2 Br\) on treatment with \(C _2 H _5 OD / C _2 H _5 O ^{-}\)gives \(Ph - CD = CH _2\) as the major product	(R) El cb reaction
(D) \(PhCH _2 CH _2 Br\) and \(PhCH _2 CH _2 Br\) react with same rate	(S) First order reaction