1.

int(sin^(3)x)/((cos^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx=

Answer»

`tan^(-1)(SECX+cosx)+C`
`log_(e)|tan^(-1)(secx+cosx)|+C`
`1/((secx+cosx)^(2))+C`
None of these

Solution :Let
`I=int_(sin^(3)x)/((COS^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx`
`impliesI=int((sin^(3)x)/(cos^(2)x))/((cos^(2)x+3+sec^(2)x)tan^(-1)(secx+cosx))dx`
`impliesI=int1/(1+(secx+cosx)^(2))XX(sinx(1-cos^(2)x))/(cos^(2)x)xx1/(tan^(-1)(secx+cosx))dx`
`impliesI=int1/(tan^(-1)(secx+cosx))xx1/(1+(secx+cosx))^(2)xx(tanx secx -sinx)dx`
`impliesI=int1/(tan^(-1)(secx+cosx))d{tan^(-1)(secx+cosx)}`
`impliesI=log|tan^(-1)(secx+cosx)|+C`


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