Question

integrate x^3/(x^2+1)^3 dx​

Answer 1

x

3

x

2

+

1

d

x

=

∫

x

2

x

2

+

1

x

d

x

=

1

2

∫

(

x

2

+

1

)

−

1

x

2

+

1

2

x

d

x

Let  

u

=

x

2

+

1

⇒

d

u

=

2

x

d

x

. Then

1

2

∫

(

x

2

+

1

)

−

1

x

2

+

1

2

x

d

x

=

1

2

∫

u

−

1

u

d

u

=

1

2

∫

(

1

−

1

u

)

d

u

=

1

2

(

u

−

ln

|

u

|

)

+

C

=

x

2

+

1

2

−

ln

(

x

2

+

1

)

2

+

C

=

x

2

2

−

ln

(

x

2

+

1

)

2

+

1

2

+

C

=

x

2

−

ln

(

x

2

+

1

)

2

+

C

(Note that as  

C

is an arbitrary constant, we can disregard the  

1

2

as we did in the last step. Adding an additional constant makes no difference when we already are considering all functions of that form which vary by a constant.)

Answer 2

Answer:

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Step-by-step explanation:

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