Evaluate the integrals \int_0^{\pi/4}\frac{\sec^2x}{(1+7\tan x)^{2/3}}ZS

beljuA 2021-10-12 Answered
Evaluate the integrals
0π4sec2x(1+7tanx)23
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Expert Answer

aprovard
Answered 2021-10-13 Author has 94 answers
To evaluate: 0π4sec2x(1+7tanx)23dx
Let substitute t=(1+7tanx)
dtdx=ddx(1+7tanx)
dtdx=7sec2xdx
dt7sec2xdx
Limit will also change accordingly.
When x0 then t1
when xπ4 then t8
Substituting the value,
0π4sec2x((1+7tanx)23)23=181(t)1(t)23dt7
=1718(t)23dt
=17[t23+123+1]18
=17[t1313]18
=37[813113}]
=37[21]
=37
Hence required answer is 37.
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