Question

Let `I_(n)=inttan^(n)xdx , n gt 1`. `I_(4)+I_(6)=atan^(5)x+bx^(5)+C` , where C is a constant of integration , then the ordered pair ( a , b) is equal to
A. `((1)/(5),-1)`
B. `(-(1)/(5),0)`
C. `(-(1)/(5),1)`
D. `((1)/(5),0)`

Answer 1

Correct Answer - d
We have , `:. I_(4)+I_(6)=inttan^(4)dx+inttan^(6)x dx`
`rArr I_(4)+I_(6)=int(tan^(4)x+tan^(6)x)dx`
`rArrI_(4)+I_(6)=inttan^(4)x(1+tan^(2)x)dx`
`rArr I_(4)+I_(6)=inttan^(4)xsec^(2)xdx =inttan^(4)xd (tanx)`
`rArrI_(4)+I_(6)=(1)/(5)tan^(5)x+C`
But, `I_(4)+I_(6)=atan^(5)x+bx^(5)+C`
`:. a=(1)/(5),b=0`
Hence , `( a ,b)=((1)/(5),0)`