x=1/2[a½+1/a½]
find x½-1/x-root x²-1
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Answer:
log(1+
2
)−
4
π
f(x)=∫
(1+x
2
)(1+
1+x
2
)
x
2
dx
Let x=tanθ
⇒dx=sec
2
θdθ=(1+x
2
)dθ
∴f(x)=∫
(1+x
2
)(1+
1+x
2
)
x
2
dx
=∫
sec
2
θ(1+secθ)
tan
2
θsec
2
θdθ
=∫
1+secθ
tan
2
θdθ
=∫
cosθ(1+cosθ)
sin
2
θdθ
=∫
cosθ(1+cosθ)
1−cos
2
θdθ
=∫
cosθ
(1−cosθ)dθ
=∫secθdθ−∫dθ
=log∣secθ+tanθ∣−θ+C
=log(x+
1+x
2
)−tan
−1
x+C
Given f(0)=0
⇒0=log1−0+C
⇒C=0
f(x)=log(x+
1+x
2
)−tan
−1
x
f(1)=log(1+
1+1
)−tan
−1
(1)
⇒f(1)=log(1+
2
)−
4
π
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