JEE Advanced · Mathematics · 27. Definite Integration

If $α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1} x}) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x$ where $\tan^{- 1} x$ takes only principal values, then the value of $(\log_{e} |1 + α| - \frac{3 π}{4})$ is

A 9
B 1
C 21
D 12

Verified Solution

Answer & Solution

Correct Answer

(A) 9

Step-by-step Solution

Detailed explanation

α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1} x}) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x

= \int_{0}^{1} (e^{9 x + 3 \tan^{- 1} x}) (9 + \frac{3}{1 + x^{2}}) d x

= (e^{9 x + 3 \tan^{- 1} x}) |_{0}^{1}

= e^{9 + \frac{3 π}{4}} - 1

\Rightarrow \ln |(1 + α)| - \frac{3 π}{4} = \ln |(1 + e^{9 + \frac{3 π}{4}} - 1)| - \frac{3 π}{4} = \ln |e^{9 + \frac{3 π}{4}}| - \frac{3 π}{4} = 9

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.

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Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.

List - I	List - II
(P) $P _2 O _3$ $+~3 H _2 O \rightarrow$	(1) $P ( O )\left( OCH _3\right)$ $Cl _2$
(Q) $P _4+3 NaOH$ $+~3 H _2 O \rightarrow$	(2) $H _3 PO _3$
(R) $PCl _5+ CH _3$ $COOH \rightarrow$	(3) $PH _3$
(S) $H _3 PO _2$ $+~2 H _2 O ~+$ $4 AgNO _3 \rightarrow$	(4) $POCL _3$
	(5) $H _3 PO _4$

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List - I	List - II
(P) \(P _2 O _3\) \(+~3 H _2 O \rightarrow\)	(1) \(P ( O )\left( OCH _3\right)\) \(Cl _2\)
(Q) \(P _4+3 NaOH\) \(+~3 H _2 O \rightarrow\)	(2) \(H _3 PO _3\)
(R) \(PCl _5+ CH _3\) \(COOH \rightarrow\)	(3) \(PH _3\)
(S) \(H _3 PO _2\) \(+~2 H _2 O ~+\) \(4 AgNO _3 \rightarrow\)	(4) \(POCL _3\)
	(5) \(H _3 PO _4\)

If α= ∫01e9x+3tan-1⁡x 12+9x21+x2 dx where tan-1⁡x takes only principal values, then the value of loge⁡1+α- 3π4 is

Step-by-step Solution

If $α = \int_{0}^{1} (e^{9 x + 3 \tan^{- 1} x}) (\frac{12 + 9 x^{2}}{1 + x^{2}}) d x$ where $\tan^{- 1} x$ takes only principal values, then the value of $(\log_{e} |1 + α| - \frac{3 π}{4})$ is