1any takers
\int \frac{t(1-t^{2})^{-7/4}}{\sqrt{t+2\sqrt{1-t^{2}}+\sqrt{t+3\sqrt{1-t^{2}}}}}dt
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2 Answers
1put t=sinx
hence dt=cosxdx
putting the values i=∫(sinxcosx-5/2dx)/(√sinx+3cosx+√sinx+2cosx)
rationalising the denominator by multiplying both sides by √sinx+3cosx-√sinx+2cosx
we get
i=∫sinxcosx-7/2(√sinx+3cosx-√sinx+2cosx)dx
separating the two
i=∫sinxcosx-7/2√sinx+3cosxdx -∫sinxcosx-7/2√sinx+2cosxdx
for each part take cosx common from the square root part
let me show only the first part
i1=∫sinxcosx-7/2√cosx(tanx+3)dx
=∫sinxcosx-3√tanx+3dx
=∫tanxsec2x√tanx+3dx
substitute tanx+3=z2
hence sec2xdx=2zdz
thus i1=2∫(z4-3z2)dz
=2{(tanx+3)5/2/5-(tanx+3)3/2}
similarly i2=2{(tanx+2)5/2/5- 2/3(tanx+2)3/2}
hence i=i1 - i2
put the value of tanx =t/√1-t2
hence get the result and reply