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:

∫$x{e}^{x}dx$

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) = $e^x$

∫$x{e}^{x}dx$

= $x{e}^x-$∫$1(e^x)dx$

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

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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