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Exercise on Integration
1.1 Substitution
Use a suitable substitution to evaluate the following integral.
dx √ 2 − 5 x
e 3 x
ex^ + 1
dx
x √ 1 − x^2
dx
x 2 3
1 + x^3 dx
xdx
(1 + x^2 )^2
dx √ x(1 + x)
x^2
sin
x
dx
xe −x^2 dx
(ln x) 2
x
dx
exdx
2 + ex
dx
ex^ + e−x
cos
x √ x
dx
tan xdx
dx
1 + ex
x(x 2
99 dx
x √ 25 − x^2
dx
x √ 3 x^2 + 1
dx
x^2 √ 9 − x^3
dx
x(x + 2) 99 dx
xdx √ 4 x + 5
x
x − 1 dx
(x + 2)
x − 1 dx
xdx √ x + 9
x 3 (1 + 3x 2 )
1 (^2) dx
1.2 Integration By Parts
ln xdx
x 2 ln xdx
ln x
x
dx
xe −x dx
x 2 e − 2 x dx
x cos xdx
x 2 sin 2xdx
(ln x) 2 dx
sin − 1 xdx
x tan − 1 xdx
ln(x +
1 + x^2 )dx
x sin 2 xdx
sin(ln x)dx
x sin 4xdx
x cos − 1 xdx
tan − 1 xdx
x 99 ln xdx
ln x
x^101
dx
x sec 2 xdx
e 2 x cos 3xdx
1.3 Reduction Formula
Prove the following reduction formulas.
- In =
x n e ax dx; In =
xneax
a
n
a
In− 1 , n ≥ 1
- In =
cos n xdx; In =
sin x cosn−^1 x
n
n − 1
n
In− 2 , n ≥ 2
- In =
sin n x
dx; In = −
cos x
(n − 1) sin n− 1 x
n − 2
n − 1
In− 2 , n ≥ 2
- In =
x n cos xdx; In = xn^ sin x + nxn−^1 cos x − n(n − 1)In− 2 , n ≥ 2
- In =
dx
(x^2 − a^2 )n^
; In = −
x
2 a^2 (n − 1)(x^2 − a^2 )n−^1
2 n − 3
2 a^2 (n − 1)
In− 1 , n ≥ 1
- In =
xndx √ x + a
; In =
2 xn
x + a
2 n + 1
2 an
2 n + 1
In− 1 , n ≥ 1
- In =
(ln x) n dx; In = x(ln x) n − nIn− 1 , n ≥ 1.
- In =
0
x n
1 − xdx; In =
2 n
2 n − 3
In− 1 , n ≥ 2.
1.6 Rational Functions
Evaluate the following integrals of rational functions.
x^2 dx
1 − x^2
x^3
3 + x
dx
(1 + x)^2
1 + x^2
dx
dx
x^2 + 2x − 3
dx
(x^2 − 2)(x^2 + 3)
x^2 + 1
(x + 1)^2 (x − 1)
, dx
x 2
(x^2 − 3 x + 2)^2
, dx
x^2 + 5x + 4
x^4 + 5x^2 + 4
, dx
dx
(x + 1)(x^2 + 1)
2 x 3 − 4 x 2 − x − 3
x^2 − 2 x − 3
dx
4 − 2 x
(x^2 + 1)(x − 1)^2
dx
dx
x(x^2 + 1)^2
x^2 dx
(x − 1)(x − 2)(x − 3)
xdx
x^2 (x^2 − 2 x + 2)
1.7 t-method
Use t-substitution to evaluate the following integrals.
dx
sin^3 x
dx
1 + sin x
dx
sin x cos^4 x
dx
2 + sin x
1 − cos x
3 + cos x
dx
cos x + 1
sin x + cos x
dx
1.8 Miscellaneous
Evaluate the following integrals.
(ln x)^2
x
dx
x(ln x) 2 dx
xdx √ 1 − x^2
x + 4
(x + 1)^2
dx
cos 3 x
sin^2 x
dx
xdx
(1 + x^2 )^2
e 2 x dx
1 + ex
dx
x(1 + 2 ln x)
cos 2 x sin 3 xdx
sin 2x
1 + cos^2 x
dx
e
1 x
x^2
dx
sin x
cos^2 x
dx
x tan 2 xdx
cot x
1 + sin x
dx
x 3 dx
x^2 − 1
dx
e^2 x^ + ex^ − 2
ln x
x
1 + ln x
dx
9 − x^2
x^2
dx
x^2 dx
x^2 + 1
dx √ x^2 + 9
cos^3 x
sin x
dx
x^2 + 8
x^2 − 5 x + 6
dx
xdx √ x − 2
dx √ 1 + ex
cos(ln x)dx
x sin 2 xdx
dx √ ex^ − 1
4 dx
x^2
4 − x^2
x + 1
x^2 (x − 1)
dx
sec 3 x tan xdx
x 3
x^2 + 1dx
cos 2x sin 3xdx
x 4
x^3 + x
dx
x 3 dx √ x^2 + 4
dx
(x^2 − 1)^2
dx
1 +
x
cos
xdx
tan 4 xdx
dx √ x(x − 1)
x 2 tan − 1 xdx
sin − 1 xdx
xdx √ 1 − x
x + 1
x
dx
x
1 − xdx
Section 1.4: Trigonometric Integrals
- − cot x 2
+ C
1 6 sin
6 x + C
1 4 sin 2x^ −^
1 16 sin 8x^ +^ C
- 3 sin x 6 +^
3 5 sin^
5 x 6 +^ C
- sin x − 1 3 sin
3 x + C
- 38 x − 14 sin 2x + 321 sin 4x + C
1 3 tan
(^3) x + C
3 sec^3 x + − sec x + C
- −x − cot x + C
1 sin x +^
1 2 ln^
1+sin x 1 −sin x +^ C
11. −^1
2 cos^2 x + 1 2 ln(1 + cos^2 x) + C
tan^4 4 −^
tan^2 x 2 −^ ln^ |^ cos^ x|^ +^ C
- −8 cot 2x − 8 3 cot
3 2 x + C
1 8 cos 4x^ −^
1 12 cos 6x^ +^ C
x 4 +^
sin 2x 8 +^
sin 4x 16 +^
sin 6x 24 +^ C
cos^8 (x) 8 −^
cos^6 (x) 6 +^ C
sin^9 (x) 9 −^
2 sin^7 (x) 7 +^
sin^5 (x) 5 +^ C
1 6 cos
(^5) x sin x + 1 24 cos
(^3) x sin x + 1 16 cos^ x^ sin^ x^ +^
1 16 x^ +^ C.
Section 1.5: Trigonometric Substitution
- x − tan − 1 x + C
- √x 1 −x^2
+ C
1 − x^2 + sin − 1 x + C
√x 1+x^2
+ C
9 2 sin
− 1 x 3 −^
x 2
9 − x^2 + C
- ln |x +
4 + x^2 | + C
16 − x^2
x^3 4 − 2 x
+32 sin−^1
x 4
+C
√ x^2 + 4 x +^ C
√ x 4 x^2 +
√x−^1 2 x−x^2
Section 1.6: Rational Functions
- −x + 1 2 ln^ |
1+x 1 −x |^ +^ C
- 9x − 3 2 x
2
1 3 x
3 − 27 ln |3 + x| + C
- x + ln(1 + x^2 ) + C
1 4 ln^ |
x− 1 x+3 |^ +^ C
1 10
√ 2 ln | x−
√ 2 x+
√ 2
1 5
√ 3 tan−^1 √x 3
+ C
1 x+1 +^
1 2 ln^ |x
2 − 1 | + C
5 x− 6 x^2 − 3 x+2 + 4 ln^ |
x− 1 x− 2 |^ +^ C
- tan−^1 x + 5 6 ln^
x^2 + x^2 +4 +^ C
- 12 tan−^1 x + 14 ln (x+1)^2 x^2 +1 +^ C
- x 2
- 2 ln |x + 1| + 3 ln |x − 3 | + C
- tan−^1 x − (^) x−^11 + ln x
(^2) + (x−1)^2 +^ C
- (^) 2(x^12 +1) + ln |x| − 12 ln(x^2 + 1) + C
- 92 ln(x−3)−4 ln(x−2)+ 12 ln(x−1)+C
- 14 ln
x^2 x^2 − 2 x+
− 12 tan−^1 (1 − x) + C
Section 1.7: t-method
- − cos^ x 2 sin^2 x
2 ln | tan x 2
| + C
- tan x − sec x + C
- (^) cos^1 x + (^) 3 cos^13 x + ln | tan x 2 | + C
√^2 3 tan − 1
2 tan(x 2 )+ √ 3
+ C
2 tan − 1
tan(x 2 ) √ 2
− x + C
2 (x + ln(sin x + cos x + 3)) − √^1 7
tan−^1
2 tan(x 2 )+ √ 7
+ C
Section 1.8: Miscellaneous
1 3 (ln^ x)
3
1 2 x
(^2) (ln x) (^2) − 1 2 x
(^2) ln x + 1 4 x
2 + C
1 − x^2 + C
- ln |x + 1| − 3 x+1 +^ C
- − 1 sin x −^ sin^ x^ +^ C
- − 1 2(1+x^2 ) +^ C
- ex^ − ln(1 + ex) + C
1 2 ln^ |1 + 2 ln^ x|^ +^ C
- 15 cos^5 x − 13 cos^3 x + C
- − ln(1 + cos 2 x) + C
- −e
1 x (^) + C
- sec x + C
- −x
2 2
- − ln |1 + csc x| + C
1 2 x
2
1 2 ln^ |x
2 − 1 | + C
x
2
1 3 ln^ |e
x (^) − 1 | + C
4 3
1 + ln x + 2 3 (ln^ x)
1 + ln x + C
√ 9 −x^2 x −^ sin
− (^1) x 3 +^ C
- x − tan − 1 x + C
- ln |x +
x^2 + 9| + C
- ln | sin x| − 1 2 sin
2 x + C
- x + 17 ln |x − 3 | − 12 ln |x − 2 | + C
2 3 (x^ −^ 2)
3 (^2) + 4(x − 2)
1 (^2) + C
- x − 2 ln(1 +
1 + ex) + C
x 2 (cos(ln^ x) + sin(ln^ x)) +^ C
- 1 4 x^2 − 1 4 x sin 2x − 1 8 cos 2x + C
- −2 sin−^1 e−^
x (^2) + C
√ 4 −x^2 x +^ C
- (^1) x − 2 ln |x| + 2 ln |x − 1 | + C
- 13 sec^3 x + C
- 13 x^2 (x^2 + 1)
3 (^2) − 2 15 (x
2 + 1)^52 + C
- − 101 cos 5x − 12 cos x + C
- 12 x^2 − ln |x| + 12 ln(x^2 + 1) + C
- 13 (x^2 + 4)
3 (^2) − 4
x^2 + C
- 14 ln |x + 1| − 14 ln |x − 1 | − (^) 2(xx (^2) −1) + C
x − 2 ln(1 +
x) + C
x sin
x + 2 cos
x + C
- 13 tan^3 x − tan x + x + C
- ln |
x − 1 | − ln |
x + 1| + C
1 3 x
(^3) tan− (^1) x − 1 6 x
6 ln(x
2 + 1) + C
- x sin − 1 x +
1 − x^2 + C
- sin−^1
x −
x
1 − x + C
x + 1 + ln |
x − 1 | − ln |
x + 1| + C
- 14 sin−^1
x − (^14)
x
1 − x(1 − 2 x) + C
