Evaluate : (i) `int cos^(3)xsinxdx` (ii) `int(sqrt(sinx))cosxdx` (iii) `int(cosec^(2)x)/((1+cotx))dx` (iv) `int(sinx)/((3+4cosx)^(2))dx`

Explanation

(i) Put cos x=t so that sin x dx =-dt.
`:.intcos^(3)xsinxdx=-intt^(3)dt=-(t^(4))/(4)+C=-(1)/(4)cos^(4)x+C`.
(ii) Put sin x=t so that cos x dx = dt.
`:.int(sqrt(sinx))cosxdx=intsqrt(t)dt=(2)/(3)t^(3//2)+C=(2)/(3)(sinx)^(3//2)+C`.
(iii) Put (1+cot x)=t so that `-cosec^(2)xdx=dt`.
`:.int("cosec"^(2)x)/((1+cotx))dx=-int(1)/(t)dt=-logt+C=-log|{:(1+cotx):}|+C`.
(iv) Put (3+4cos x) =t so that -4sin x dx=dt.
`:.int(sinx)/((3+4cosx)^(2))dx=-(1)/(4)int(1)/(t^(2))dt=(1)/(4t)+C=(1)/(4(3+4cosx))+C`.