COMEDK2015MathematicsDefinite Integration
The value of the integral ₀^ x ^ 2 n x ^ 2 n x+ ^ 2 n x d x is
Options
- A²
- B2 ²
- C² 4
- D² 2
Correct answer
C. ² 4
Step-by-step solution
Let I= ₀^ x ^ 2 n x ^ 2 n x+ ^ 2 n x d x ...(i) I= ₀^ ( -x) ^ 2 n ( -x) ^ 2 n ( -x)+ ^ 2 n ( -x) d x = ₀^ ( -x) ^ 2 n x ^ 2 n x+ ^ 2 n x d x ...(ii) Adding Eqs. (i) and (ii), we get 2 I= ₀^ ^ 2 n x ^ 2 n x+ ^ 2 n x d x ...(iii) Let I₁= ₀^ ^ 2 n x ^ 2 n x+ ^ 2 n x d x I₁=2 ₀^ / 2 ^ 2 n x ^ 2 n x+ ^ 2 n x d x ...(iv) I₁=2 ₀^ / 2 ^ 2 n ( 2 -x ) ^ 2 n ( 2 -x )+ ^ 2 n ( 2 -x ) d x =2 ₀^ / 2 ^ 2 n x ^ 2 n x+ ^ 2 n x d x ...(v) Adding Eqs. (iv) and (v), we get aligned &2 I₁=2 ₀^ / 2 1 d x=2 2 = & I₁= 2 aligned Substitutin