integrate 1/(sqrt(1 + x) + sqrt(x)) dx from 0 to 1
Answers
Step-by-step explanation:
Explanation:
We can evaluate this integral using integration by substitution, or u-substitution. We pick some part of the integrand to set equal to some variable (such as
u
, but any variable is an option). Good places to look at first include under a radical or in the denominator. This is not always the case, but it is in this one.
We can set
u
=
1
−
x
Therefore,
d
u
=
−
1
d
x
−
d
u
=
d
x
We can substitute these values into our integral. We get:
−
∫
1
√
u
d
u
Which we can rewrite as:
−
∫
u
−
1
2
d
u
Integrating, we get:
−
2
u
1
2
From here you have two options on evaluating for the given limits of integration. You can either choose now to substitute
1
−
x
back in for
u
and evaluate from 0 to 1, or you can change the limits of integration and evaluate with u. I will demonstrate both options.
Substituting
1
−
x
back in for
u
,
−
2
(
1
−
x
)
1
2
−
2
[
(
1
−
1
)
1
2
−
(
1
−
0
)
1
2
]
−
2
(
−
1
)
Final answer: 2
Changing limits of integration:
u
=
1
−
x
u
=
1
−
(
1
)
u
=
0
(new upper limit)
u
=
1
−
0
u
=
1
(new lower limit)
Evaluating, we have
−
2
[
(
0
)
1
2
−
(
1
)
1
2
]
−
2
(
−
1
)
Final answer: 2