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KEAM 2013 Math Paper
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© examsnet.com
Question : 94
Total: 120
If
u
=
t
a
n
−
1
(
√
1
−
x
2
−
1
x
)
and
v
=
s
i
n
−
1
x
,
then
d
u
d
v
=
√
1
−
x
2
−
1
2
1
−
x
−
2
Validate
Solution:
Given,
u
=
tan
−
1
{
√
1
−
x
2
−
1
γ
}
and
v
=
sin
−
1
x
Put
x
=
sin
θ
⇒
θ
=
sin
−
1
x
,
then
u
=
tan
−
1
{
cos
θ
−
1
sin
θ
}
(
∵
1
−
cos
2
θ
=
sin
2
θ
)
=
−
tan
−
1
{
1
−
cos
θ
sin
θ
}
=
−
tan
−
1
{
2
sin
2
θ
∕
2
2
sin
θ
∕
2
⋅
cos
θ
∕
2
}
=
−
tan
−
1
(
tan
θ
2
)
=
−
1
2
θ
=
−
1
2
sin
−
1
x
∴
d
u
d
x
=
−
1
2
√
1
−
x
2
and
d
v
d
x
=
1
√
1
−
x
2
∴
d
u
d
v
=
d
u
d
x
×
d
x
d
v
=
−
1
2
√
1
−
x
2
×
√
1
−
x
2
=
−
1
2
© examsnet.com
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