The value of the definite integral $\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{(1 + {\rm{ }}{e^{x\,\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $ is equal to

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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The value of the definite integral

\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{(1 + \mathrm{ }{e^{x\,\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}

is equal to

View answer and explanation

Correct option: B

\frac{\pi }{{2\sqrt 2 }}

Option B (

\frac{\pi }{{2\sqrt 2 }}

) is correct because it is the result of evaluating the definite integral after applying properties of definite integrals and trigonometric substitutions.

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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}

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

equals The value of

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[*]

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\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

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

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