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Lecture 19 Integration

The document discusses integration techniques, specifically integration by parts, and provides various applications and examples of integrals involving polynomials, trigonometric, logarithmic, and hyperbolic functions. It also includes reduction formulas for sine, cosine, and hyperbolic functions, demonstrating how to derive integrals recursively. Additionally, it presents specific integrals and their evaluations using these techniques.

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quocphuongtri.pham.csps
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7 views16 pages

Lecture 19 Integration

The document discusses integration techniques, specifically integration by parts, and provides various applications and examples of integrals involving polynomials, trigonometric, logarithmic, and hyperbolic functions. It also includes reduction formulas for sine, cosine, and hyperbolic functions, demonstrating how to derive integrals recursively. Additionally, it presents specific integrals and their evaluations using these techniques.

Uploaded by

quocphuongtri.pham.csps
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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12

Integration by parts
Z Z
dv du
u dx = uv − v dx
dx dx
Z Z
uv 0 (x)dx = uv − vu0 (x)dx

• Applications:
Z
Pn (x) sin(ax + b)dx Let u = Pn (x)
Z
Pn (x) cos(ax + b)dx Let u = Pn (x)
Z
Pn (x)eax+b dx Let u = Pn (x)
Z
Pn (x)(ln(ax + b))m dx Let u = (ln(ax + b))m
Z
eax+b sin(mx + n)dx Let u = eax+b or u = sin(mx + n)
Z
eax+b cos(mx + n)dx Let u = eax+b or u = cos(mx + n)
Z Z
Pn (x) arcsin(ax + b)dx = Pn (x) sin−1 (ax + b)dx Let u = sin−1 (ax + b)
Z Z
Pn (x) arctan(ax + b)dx = Pn (x) tan−1 (ax + b)dx Let u = tan−1 (ax + b)
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• Example.
(1) R sinh−1 xdx
R

(2) R cosh−1 xdx


(3) R tanh−1 xdx
(4) R tan−1 xdx
(5) R sin−1 xdx
(6) R x3 arctan xdx =R x3 tan−1 xdx
R

(7) R x3 arccotxdx = x3 cot−1 xdx


(8) R x2 sin 2xdx
(9) 3x
Re 3xsin 2xdx
(10) e cos 2xdx
R √x
(11) e dx √
R
(12) ln(x + x2 + 1)dx
R ln(1 + x2 )
(13) dx
R 2 x2
(14) R x ln xdx
(15) x3 ln2 xdx
R ln x
(16) dx
R x22 3x
(17) R x e dx
(18) x2 e−2x dx
14

INTEGRATION TECHNIQUES
Z
dx
(1) √ , for a > 0,
a2 − x2
Z
dx
(2) √ ,
9 − 4x2
Z
dx
(3) √ ,
x − a2
2
Z
dx
(4) ,
a + x2
2
Z
dx
(5) 2
,
x − 3x + 7
Z
dx
(6) √ ,
x + a2
2
Z 1.5 p
(7) 9 − x2 dx,
0
Z 1.5 p
(8) 9 − x2 dx,
0
Z
dx
(9) , by using a hyperbolic substitution,
x2
+ 4x
Z
dx
(10) ,
sinh x
Z
(11) cosh−1 xdx,
Z
(12) sinh−1 xdx,
Z
(13) tanh−1 xdx,
Z
(14) cos−1 xdx,
Z
(15) sin−1 xdx,
Z
(16) tan−1 xdx.
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Reduction formulae
sin3 xdx, find the result for I3 using the integration by parts.
R
• Example. Given that I3 =
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Reduction formulae
cosn xdx. Prove that for any n ≥ 2,
R
(1) Let In =
1 n−1
In = cosn−1 x sin x + In−2 .
n n
Determine I0 , I1R, I2 , I3 , I4 .
(2) Let In = sinn xdx. Prove that for any n ≥ 2,
1 n−1
In = − cos x sinn−1 x + In−2 .
n n
Determine I0 , I1R, I2 , I3 , I4 .
(3) Let In = coshn 2xdx. Prove that for any n ≥ 2,
1 n−1
In = sinh 2x coshn−1 2x + In−2 .
2n n
(4) Let In = cothn xdx. Prove that for any n ≥ 2,
R

1
In = − cothn−1 x + In−2 .
n−1
(5) Let In = xn ex dx. Prove that for any n ≥ 1,
R

In = xn ex − nIn−1 .
x2 (1 + x6 )n dx. Prove that for any n ≥ 1,
R
(6) Let In =
x3 (1 + x6 )n 2n
In = + In−1 .
3(2n + 1) 2n + 1
xn (x2 − 1)9 dx. Prove that for any n ≥ 2,
R
(7) Let In =
xn−1 (x2 − 1)10 n−1
In = + In−2 .
n + 19 n + 19
R π/6
(8) Let In = 0 tann dx. Prove that for any n ≥ 2,
√ !n−1
1 3
In = − In−2 .
n−1 3
Then, find I4 .
R π/2 R π/2
(9) Wallis’ integral: In = 0 sinn dx = 0 cosn dx. Prove that for any n ≥ 0,
n+1
In+2 = In .
n+2
Then, find I9 and I10 .
R π/2
(10) Let In = 0 xn cos xdx. Prove that for any n ≥ 2,
 π n
In = − n(n − 1)In−2 .
2
Then, find I3 .
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