Integration Techniques' Examples

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Antidifferentiation

Example 1 :
Question:

Find the antiderivative for the following function.

f(x) = $e^{3x}$

Solution:

∫ f(x)dx

∫$e^{3x}$dx

=${\frac {1}{3}}e^{3x}$

=${\frac {e^{3x}}{3}}$ + C

Example 2 :
Question:

Find the antiderivative for the following function.

f(x) = -4x

Solution:

∫f(x)dx

∫ -4x dx

=$-2x^{2}$ + C

Example 3 :
Question:

∫4$x^{3}$dx

Solution:

= $x^{4}$ + C

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Integration By Substitution

Example 1 :
Question:

Determine:

∫${(x^{2}+1)}^{3}2xdx$

Solution:

Set:

$u=x^{2}+1$

$du={\frac {d}{dx}}(x^{2}+1)dx$

$du=2xdx$

∫${(x^{2}+1)}^{3}2xdx$

∫$=u^{3}du$

= ${\frac {1}{4}}u^{4}+C$

= ${\frac {1}{4}}{(x^{2}+1)}^{4}+C$

Example 2 :
Question:

Determine:

∫${\frac {({lnx}^{2})}{x}}dx$

Solution:

Set:

$u=lnx$

$du={\frac {1}{x}}dx$

∫${\frac {({lnx}^{2})}{x}}dx$

∫$=({lnx}^{2}){\frac {1}{x}}dx$

= ∫$u^{2}du$

= ${\frac {u^{3}}{3}}+C$

= ${\frac {(lnx)^{3}}{3}}+C$

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Integration By Parts

Example 1 :
Question:

∫$xe^{x}dx$

Solution:

∫ f(x)g(x)dx = f(x)G(x) - ∫f'(x)G(x)dx

Set:

f(x) = x

f'(x) = 1

g(x) = $e^{x}$

G(x) = $e^{x}$

∫$xe^{x}dx$

= $xe^{x}-$∫$1(e^{x})dx$

= $xe^{x}-e^{x}+C$

Example 2 :
Question:

∫$x(x+5)^{8}dx$

Solution:

∫ f(x)g(x)dx = f(x)G(x) - ∫f'(x)G(x)dx

Set:

f(x) = x

f'(x) = 1

g(x) = $(x+5)^{8}$

G(x) = ${\frac {1}{9}}(x+5)^{9}$

∫$x(x+5)^{8}dx$

= $x{\frac {1}{9}}(x+5)^{9}-$ ∫$1{\frac {1}{9}}(x+5)^{9}dx$

= ${\frac {1}{9}}x(x+5)^{9}-{\frac {1}{9}}$∫$(x+5)^{9}dx$

= ${\frac {1}{9}}x(x+5)^{9}-{\frac {1}{9}}{\frac {1}{10}}(x+5)^{10}+C$

= ${\frac {1}{9}}x(x+5)^{9}-{\frac {1}{90}}(x+5)^{10}+C$

Example 3 :
Question:

∫$xsinxdx$

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

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