Math, asked by dnyanudhande, 11 months ago

evaluate 1+x+√x+x^2/√x+√1+x.dx​

Answers

Answered by abhinav252525
0

Answer:

d

x

x

+

x

2

+

x

+

1

=

(

x

2

+

x

+

1

x

)

+

ln

|

x

+

1

|

1

2

ln

2

x

+

1

+

2

x

2

+

x

+

1

+

ln

(

2

3

)

(

3

+

2

x

2

+

x

+

1

)

+

(

2

x

+

1

)

(

2

3

)

(

3

+

2

x

2

+

x

+

1

)

(

2

x

+

1

)

Explanation:

Let

I

=

d

x

x

+

x

2

+

x

+

1

Rationalize:

I

=

d

x

x

+

x

2

+

x

+

1

x

x

2

+

x

+

1

x

x

2

+

x

+

1

Simplify:

I

=

x

2

+

x

+

1

x

x

+

1

d

x

Rearrange:

I

=

x

2

+

x

+

1

x

+

1

d

x

(

1

1

x

+

1

)

d

x

Complete the square in the square root:

I

=

1

2

(

2

x

+

1

)

2

+

3

x

+

1

d

x

(

x

ln

|

x

+

1

|

)

Apply the substitution

2

x

+

1

=

3

tan

θ

:

I

=

1

2

3

sec

θ

3

2

tan

θ

+

1

2

(

3

2

sec

2

θ

d

θ

)

x

+

ln

|

x

+

1

|

Simplify:

I

=

3

2

sec

2

θ

3

tan

θ

+

1

sec

θ

d

θ

x

+

ln

|

x

+

1

|

Since

sec

2

θ

=

tan

2

θ

+

1

:

I

=

1

2

(

3

tan

θ

1

+

4

3

tan

θ

+

1

)

sec

θ

d

θ

x

+

ln

|

x

+

1

|

Rearrange:

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