`int (dx)/((4sin^(2)x+5cos^(2)x))=?`

1.	`int (dx)/((4sin^(2)x+5cos^(2)x))=?`
Answer» On dividing the numerator and denominator by `cos^(2)x` , we get `int(dx)/((4sin^(2)x+5cos^(2)x))=int(sec^(2)x)/((4tan^(2)x+5))dx` `=int(dt)/(4t^(2)+5)` [ putting tan x = t] `=(1)/(4)int(dt)/((t^(2)+(5)/(4)))=(1)/(4)int(dt)/([t^(2)+(sqrt(5)/(2))^(2)])` `=(1)/(4)(1)/((sqrt(5)/(2)))tan^(-1)(t)/((sqrt(5)/(2)))+C` `=(1)/(2sqrt(5))tan^(-1)((2t)/(sqrt(5)))+C=(1)/(2sqrt(5))tan^(-1)((2tanx)/(sqrt(5)))+C`.

Discussion

`int (dx)/((4sin^(2)x+5cos^(2)x))=?`
Evaluate `int(sqrt(tanx)+sqrt(cotx))dx`.
Evaluate `intsqrt(cotx)dx`.
`int(x^(2))/((x^(2)+6x-3))dx`
`int(dx)/((1+cos^(2)x))`
`int(dx)/((2+sin^(2)x))`
`int (dx)/((cos ^(2)x-3sin^(2)x))=?`
Evaluate `int(x)/((x^(4)-x^(2)+1))dx`.
Evaluate `int((2x+1))/((4-3x-x^(2)))dx`.
Evaluate: `int1/(a^2sin^2x+b^2cos^2x) dx`