Math, asked by soni0701, 2 months ago

integrate 1/(sqrt(1 + x) + sqrt(x)) dx from 0 to 1​

Answers

Answered by villian9120
0

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

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