Integration Techniques' Examples

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Antidifferentiation

Example 1 :
Question:

Find the antiderivative for the following function.
f(x) =
Solution:
∫ f(x)dx
dx
=
= + C


Example 2 :
Question:

Find the antiderivative for the following function.
f(x) = -4x
Solution:
∫f(x)dx
∫ -4x dx
= + C


Example 3 :
Question:

∫4dx
Solution:
= + C

Integration By Substitution

Example 1 :
Question:

Determine:
Solution:
Set:
=
=


Example 2 :
Question:

Determine:
Solution:
Set:
= ∫
=
=



Integration By Parts

Example 1 :
Question:

Solution:
∫ f(x)g(x)dx = f(x)G(x) - ∫f'(x)G(x)dx
Set:
f(x) = x
f'(x) = 1
g(x) =
G(x) =
=
=


Example 2 :
Question:

Solution:
∫ f(x)g(x)dx = f(x)G(x) - ∫f'(x)G(x)dx
Set:
f(x) = x
f'(x) = 1
g(x) =
G(x) =
=
=
=
=


Example 3 :
Question:

Solution:
∫ f(x)g(x)dx = f(x)G(x) - ∫f'(x)G(x)dx
Set:
f(x) = x
f'(x) = 1
g(x) = sinx
G(x) = -cosx
∫xsinxdx
= -xcosx - ∫1(-cosx)dx
= -xcosx + ∫cosxdx
= -xcosx + sinx + C