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JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1
∫  1  +  x2x   1  +  x4  dx\int \,\,\frac{{1\,\, + \,\,{x^2}}}{{x\;\,\sqrt {1\,\, + \,\,{x^4}} }}\,\,dx is equal to
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Correct option: B
log⁡  ∣x  −  1x  +  (x  −  1x)2  +  2∣  +  C\log \,\,\left| {x\,\, - \,\,\frac{1}{x}\,\, + \,\,\sqrt {{{\left( {x\,\, - \,\,\frac{1}{x}} \right)}^2}\,\, + \,\,2} } \right|\,\, + \,\,C
Option B is correct. The integral simplifies by dividing numerator and denominator by x2 and then substituting to get a standard logarithmic form.

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∫0∞  log⁡  (x+1x)  dx1+x2\int \nolimits_0^\infty \,\,\log \,\,\left( {x + \frac{1}{x}} \right)\,\,\frac{{dx}}{{1 + {x^2}}} is equal to If f(x)f\left( x \right) is differentiable and ∫0t2  x   f(x)  dx=25  t5\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}, then f(425)f\left( {\frac{4}{{25}}} \right) equals The value of ∫011[x]3.dx\int\limits_{{0}}^{11}[x]^{3} . \mathrm{dx}, where [∗][*] denotes the greatest integer function, is The integral
∫24log⁡x2log⁡x2+log⁡(36−12x+x2)dx\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
∫−20(x3+3x2+3x+3+(x+1)cos⁡(x+1))dx\int\limits_{{-2}}^0\left(x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right) d x
equals If I  (m,  n)=∫01  tm  (1+t)n  dtI\,\,\left( {m,\,\,n} \right) = \int \nolimits_0^1 \,\,{t^m}\,\,{\left( {1 + t} \right)^n}\,\,dt,then the expression for I(m,  n)I\left( {m,\,\,n} \right) in terms of I  (m+1,  n+1)I\,\,\left( {m + 1,\,\,n + 1} \right) is
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