1.

Evaluate (i) int((2x+5))/((x^(2)+5x+9))dx (ii) int((6x-7))/((3x^(2)-7x+5))dx (iii) int((cosx-sinx))/((cosx+sinx))dx (iv) int(sec x)/(log(secx+tanx))dx

Answer»

Solution :(i) PUT `(x^(2)+5x+9)` =t so that (2x+5)dx=DT.
`:.int((2x+5))/((x^(2)+5x+9))dx=int(1)/(t)dt=log|t|+C`
`=log|{:x^(2)+5c+9:}|+C`.
(ii) Put (cosx+sinx)=t so that (cosx-sinx)dx=dt.
`:.int((cosx-sinx))/((cosx+sinx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(cosx+sinx):}|+C`.
(iii) Put (cosx-sinx) =t so that (cosx-sinx) dx =dt.
`:.int((cosx-sinx))/((cosx+sinx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(cosx+sinx):}|+C`
(iv) Put log(SECX+tanx)=t.
Then, on DIFFERENTIATION, we get
`(1)/((secx+tanx))*(secxtanx+sec^(2)x)dx=dt`
or sec x dx = dt.
`:.int(secx)/(log(secx+tanx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(secx+tanx):}|+C`.


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