If \int \binom{2^{\frac{1}{x}}}/{x^{2}} \dx=k \cdot ...

CBSE Class 12 Math 2025 All Sets Solved Paper

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Question : 4 of 20

Marks: +1, -0

If

∫ \;{2^{{1}/{x}}}/{x^{2}} \dx=k ⋅ 2^{\;{1}/{x}}+C,

then k is equal to

$\;{-1}/{\log 2}$
$-\log 2$
-1
$\;{1}/{2}$

Solution:

∫ \;{2^{{1}/{x}}}/{x^{2}} \dx=K ⋅ 2^{\;{1}/{x}}+C

Let

\;{1}/{x}=t

Differentiating both sides

\;{-1}/{x^{2}} \dx=d t

Putting

\;{1}/{x}=t

and

\;{d x}/{x^{2}}=-d t

⇒ ∫ \;{2^{{1}/{x}}}/{x^{2}} \dx =∫ 2^{t}(-d t)

=-∫ 2^{t} \dt

=\;{-2^{t}}/{ln\ 2}+C

=\;{-2^{{1}/{x}}}/{ln\ 2}+C

⇒ k=\;{-1}/{ln\ 2} \text " or " \;{-1}/{\log 2}

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