1.

Integrate: x^2-4/(x^2+1)(x^2+2)(x^2+3) dx

Answer»

Answer:

Explanation:

Our GOAL should be to make this mirror the arctangent INTEGRAL:

∫

1

u

2

+

1

d

u

=

arctan

(

u

)

+

C

To get the

1

in the denominator, start by factoring:

∫

1

x

2

+

4

d

x

=

∫

1

4

(

x

2

4

+

1

)

d

x

=

1

4

∫

1

x

2

4

+

1

d

x

Note that we want

u

2

=

x

2

4

, so we let

u

=

x

2

, which IMPLIES that

d

u

=

1

2

d

x

.

1

4

∫

1

x

2

4

+

1

d

x

=

1

2

∫

1

2

(

x

2

)

2

+

1

d

x

=

1

2

∫

1

u

2

+

1

d

u

This is the arctangent integral:

1

2

∫

1

u

2

+

1

d

u

=

1

2

arctan

(

u

)

+

C

=

1

2

arctan

(

x

2

)

+

C


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