JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Analyzing +4 | -1

Let

g(t)=\int\limits_{-\pi / 2}^{\pi / 2} \cos \left(\frac{\pi}{4} t+f(x)\right) d x

, where

f(x)=\log _e\left(x+\sqrt{x^2+1}\right), x \in R

. Then which one of the following is correct?

View answer and explanation

Correct option: D

\sqrt{2} g(1)=g(0)

Option D (√2 g(1)=g(0)) is correct because it is derived from the properties of the integral and the even/odd nature of the functions involved, leading to the relationship between g(1) and g(0).

More Questions from DEFINITE INTEGRATION

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\int \nolimits_0^\infty \,\,\log \,\,\left( {x + \frac{1}{x}} \right)\,\,\frac{{dx}}{{1 + {x^2}}}

is equal to If

f\left( x \right)

is differentiable and

\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}

, then

f\left( {\frac{4}{{25}}} \right)

equals The value of

\int\limits_{{0}}^{11}[x]^{3} . \mathrm{dx}

, where

[*]

denotes the greatest integer function, is The integral

\int\limits_{2}^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} d x

is equal to: The definite integral

\int\limits_{{-2}}^0\left(x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right) d x

equals If

I\,\,\left( {m,\,\,n} \right) = \int \nolimits_0^1 \,\,{t^m}\,\,{\left( {1 + t} \right)^n}\,\,dt

,then the expression for

I\left( {m,\,\,n} \right)

in terms of

I\,\,\left( {m + 1,\,\,n + 1} \right)