Question

Let (a, b) be the point of intersection of the curve \(x^{2}=2 y\) and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral \(\mathrm{I}=\int_\limits{\mathrm{a}}^{\mathrm{b}} \frac{9 \mathrm{x}^{2}}{1+5^{\mathrm{x}}} \mathrm{dx}\) is equal to :

(1) 24

(2) 27

(3) 18

(4) 21  

Answer 1

Correct option is: (1) 24  

\( x^2=2 y \text { and } y=2 x+6 \)

\( \Rightarrow x^2=4 x+12 \Rightarrow x^2-4 x+12=0 \)

\(\Rightarrow(x-6)(x+2)=0 \Rightarrow x=6 \quad x=-2\)   

in second quadrant (-2, 2)

\(\Rightarrow \int_\limits{-2}^2 \frac{9 x^2}{1+5^x} d x=5\)   ...(1)

Taking \(I=\int_\limits{-2}^2 \frac{5^x x^2}{1+5^x} \mathrm{dx}\)   ...(2)

Add (1) + (2)

\( \Rightarrow 2 \mathrm{I}=\int_\limits{-2}^2 9 \mathrm{x}^2 \mathrm{dx}=\left.3 \mathrm{x}^3\right|_{-2} ^2=48 \)

\(\mathrm{I}=24\)