JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Easy

1 of 1

Analyzing +4 | -1

Let

a, b, c

be non-zero real numbers such that

\int\limits_{{0}}^{1}\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x= \int\limits_{0}^2\left(1+\cos ^8 x\right) \times\left(a x^2+b x+c\right) d x

Then the quadratic equation

a x^2+b x+c=0

has

View answer and explanation

Correct option: B

at least one root in

(0,2)

Option B (at least one root in

(0,2)

) is correct because by applying Rolle's theorem to the integral, we can conclude that there exists at least one root in the interval (1,2), which is within (0,2).

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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)