Definite IntegrationmediumFree

∫ f(xⁿ+x⁻ⁿ)·lnx·dx/x from 0 to ∞ | JEE

JEE Maths question with a full step-by-step solution.

Question

\displaystyle\int_{0}^{\infty}f\!\big(x^{n}+x^{-n}\big)\ln x\,\dfrac{dx}{x}

equals

0

correct

f\!\big(x^{n}+x^{-n}\big)

Ca function of

x

Dcannot be determined

Solution

Step 1: Substitute

t=\ln x

, so

x=e^{t}

and

\dfrac{dx}{x}=dt

. The limits map as

x\to 0^{+}\Rightarrow t\to-\infty

and

x\to\infty\Rightarrow t\to\infty

I=\int_{-\infty}^{\infty}f\!\big(e^{nt}+e^{-nt}\big)\,t\,dt.

Step 2: Examine the parity of the integrand. The argument

e^{nt}+e^{-nt}=2\cosh(nt)

is an even function of

t

, so

f\!\big(e^{nt}+e^{-nt}\big)

is even in

t

. Step 3: Multiplying an even function by

t

(odd) produces an odd integrand:

g(t)=f\!\big(e^{nt}+e^{-nt}\big)\,t,\qquad g(-t)=-g(t).

Step 4: The integral of an odd function over the symmetric interval

(-\infty,\infty)

is zero:

I=\int_{-\infty}^{\infty}g(t)\,dt=0.

Correct answer: (1)

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