If I_n = int_0^(pi/2) (sin^2 nx)/(sin^2 x) dx , then
Prove that I_(1),I_(2),I_(3)"..." form an AP, if (i) I_(n)=int_(0)^(pi)(sin2nx)/(sinx)dx (ii) I_(n)=int_(0)^(pi)((sinnx)/(sinx))^(2)dx .
Knowledge Check
Evaluation of definite integrals by subsitiution and properties of its : I_(1)=int_(a)^(pi-a)xf(sinx)dx,I_(2)=int_(a)^(pi-a)f(sinx)dx then I_(2)=……..
A
`(pi)/(2)I_(1)`
B
`piI_(1)`
C
`(2)/(pi)I_(1)`
D
`2I_(1)`
Fundamental theorem of definite integral : I=int_(0)^(1)(sinx)/(sqrtx)dx and J=int_(0)^(1)(cosx)/(sqrtx)dx then which of the following statement is true ?
A
`Igt(2)/(3) and Jgt2`
B
`Ilt(2)/(3) and Jlt2`
C
`Ilt(2)/(3)and Jgt2`
D
`Igt(2)/(3) and Jlt2`
Method of integration by parts : I_(1)=int sin^(-1)x dx and I_(2)= int sin^(-1) sqrt(1-x^(2))dx then.....
A
`I_(1)=I_(2)`
B
`I_(2)=(pi)/(2I_(1))`
C
`I_(1)+I_(2)=(pi)/(2x)`
D
`I_(1)+I_(2)=(pi)/(2)`
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