If n (2 n +1) ∫ limits01(1- x n )2 n dx =1177 ∫ limits01(1- x n )2 n +1 dx text , then n ∈ N is equal to

Question

If n (2 n +1) ∫ limits01(1- x n )2 n dx =1177 ∫ limits01(1- x n )2 n +1 dx text , then n ∈ N is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let I 1=∫ limits01(1- x n )2 n dx, I2=∫ limits01(1-xn)2 n+1 d x I2=∫ limits01(1-xn)2 n+1 ⋅ 1 d x =.(1-xn)2 n+1 ⋅ x|0 1-∫ limits01(2 n+1)(1-xn)2 n(-n xn-1) x d x I2=-n(2 n+1) I2-I1 (2 n2+n+1) I2=n(2 n+1) I1 (I1/I2)=(2 n2+ n +1/ n (2 n +1))=(1177/ n (2 n +1)) ⇒ 2 n 2+ n -1176=0 ⇒ n =24

Q. If n(2n+1)0∫1​(1−xn)2ndx=11770∫1​(1−xn)2n+1dx, then n∈N is equal to _____

Let I1​=0∫1​(1−xn)2ndx, I2​=0∫1​(1−xn)2n+1dx I2​=0∫1​(1−xn)2n+1⋅1dx =(1−xn)2n+1⋅x∣∣​01​−0∫1​(2n+1)(1−xn)2n(−nxn−1)xdx I2​=−n(2n+1){I2​−I1​} (2n2+n+1)I2​=n(2n+1)I1​ I2​I1​​=n(2n+1)2n2+n+1​=n(2n+1)1177​ ⇒2n2+n−1176=0⇒n=24

Q. If $n (2 n + 1) 0 \int 1 (1 - x^{n})^{2 n} d x = 1177 0 \int 1 (1 - x^{n})^{2 n + 1} d x,$ then $n \in N$ is equal to _____