Make an appropriate u-substitution, and then use the Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution) and then confirm that the result is equivalent to that in part (a). Given: ∫ x ln(2 + x2)dx
Make an appropriate u-substitution, and then use the Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution) and then confirm that the result is equivalent to that in part (a). Given: ∫ x ln(2 + x2)dx
Make an appropriate u-substitution, and then use the Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution) and then confirm that the result is equivalent to that in part (a). Given: ∫ x ln(2 + x2)dx
(a) Make an appropriate u-substitution, and then use the Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution) and then confirm that the result is equivalent to that in part (a).
Given:
∫ x ln(2 + x2)dx
Transcribed Image Text:PRODUCTS OF TRIGONOMETRIC FUNCTIONS
sin(m + n)u
sin(m – n)u
+C
2(m – n)
cos(m - п)и
+C
cos(m + n)u
38.
sin mu sin nu du
40.
sin mu cos nu du
2(m + n)
2(m + n)
-u cos"+!
2(m – n)
sin(m +n)u , sin(m – n)u
+C
39. cos mu cos nu du:
| sin" u cos" u du
sin
m -1
- | sin"-² u cos" u du
41.
2(m + n)
2(т — п)
m + n
m +n
+! u cos"-! u
n-1
- sin" u cos"-² u du
m +n
m +n
PRODUCTS OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS
42. " sin bu du :
a² + b²
(a sin bu – b cos bu) + C
43.
a" cos bu du =
a² + b²
(a cos bu + b sin bu) + C
POWERS OF u MULTIPLYING OR DIVIDING BASIC FUNCTIONS
51. ue" du = e" (u – 1) + C
Swe du =we
53. fu'a" du =
44. u sin u du = sin u – u cos u +C
Suco
u cos u du = cos u +u sin u + C
45.
52.
u"a"
|u° sin u du = 2u sin u + (2 – u²) cos u + C
u"-'a" du + C
In a
46.
In a
47.
u cos u du = 2u cos u + (u² – 2) sin u +C
e du
e du
54.
(n – 1)u“-1
n -
a" du
In a
a" du
* / u' cos u du
a"
is
n-1
48.
u" sin u du = -u" cos u + n
55.
(n – 1)un-I
du
In |In u| +C
u"
u" cos u du = u" sin u – n | u"-1 sin u du
56.u In u
49.
50. fu" lau
«" In u du
(n + 1)z [(n + 1) In u – 1] + C
POLYNOMIALS MULTIPLYING BASIC FUNCTIONS
57.
du
P'(u)e
[signs alternate: + - + -..]
|p(u) sin au du = -- p(u) cos au +p'(u) sinau +
| P(u) cos au du = - p(u) sin au + p'(u) cos au -
P" (u) cos au
[signs alternate in pairs after first term: ++-- ++--...]
58.
P"(u) sin au –
[signs alternate in pairs: ++ -- ++--..]
59.
Transcribed Image Text:TABLE OF INTEGRALS
BASIC FUNCTIONS
Sa" du = na
1.
du =
n+ 1
10.
+C
du
|" = In lu| +C
In u du %3D u In u —и+С
2.
11.
|" du – " +C
12. cot u du = In |sin u| + C
3.
| sec u du = In |sec u + tan u| + C
= In (tan (7 + ju) | +c
4.
sin u du = - cos u + C
13.
5.
cos u du = sin u +C
| csc u du = In |csc u – cot u| +C
= In |tan {u| + C
14.
6.
tan u du = In |sec u|+C
lu = u sin=' u + V1 – u² +C
cot-' u du = u cot-' u + In ī+u² +C_
15.
cos- u du = u cos¯-lu – V1 - u² +C
sec-" u du = u sec-u – In \u + vu² – 1| +C
8.
16.
tanu du = u tan-u – In V1 + u² + C
- | csc-" u du = u csc-" u + In [u + /u? – 1+C
17.
9.
RECIPROCALS OF BASIC FUNCTIONS
18.
du = tan u F sec u +C
22.
du = }(u # In [sin u ± cos u|) + C
1± sinu
1t cot u
19.
du = - cot u ± csc u + C
23.
du = u + cot u F csc u + C
1t cos u
1t sec u
20. Tt tan u
du = }(u± In |cos u± sin u|) + C
24. It csc u
du = u – tan u ± sec u + C
21.
du = In [tan u| + C
du = u – In(1 ± e") + C
25.
sin u cos u
POWERS OF TRIGONOMETRIC FUNCTIONS
· | sin² u du = }u – } sin 2u + C
| cot? u du = - cot u – u + C
26.
32.
27. | cos² u du = ļu + į sin 2u + C
33. |
sec²
= tan u + C
28. | tan? u du = tan u – u +C
34. csc² u du = – cot u + C
1
sin" u du = -- sin"
u cos u + "=' | sin"-2 u du
cot" u du :
cot"-1
: / cor"-2,
29.
35.
и du
n- 1
30. f
n -
u sin u +
cos"-2 u du
п — 2
i/ sec"-2
cos" u du = - cos"
36.
sec" u du =
sec"-2 u tan u +
и du
n- 1
n - 2
tan" u du =
| csc" u du
csc"-2 u cot u +
| csc"-2 u du
31.
tan"
и du
37.
tan
n- 1
n-1
n- 1
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![PRODUCTS OF TRIGONOMETRIC FUNCTIONS
sin(m + n)u
sin(m – n)u
+C
2(m – n)
cos(m - п)и
+C
cos(m + n)u
38.
sin mu sin nu du
40.
sin mu cos nu du
2(m + n)
2(m + n)
-u cos"+!
2(m – n)
sin(m +n)u , sin(m – n)u
+C
39. cos mu cos nu du:
| sin" u cos" u du
sin
m -1
- | sin"-² u cos" u du
41.
2(m + n)
2(т — п)
m + n
m +n
+! u cos"-! u
n-1
- sin" u cos"-² u du
m +n
m +n
PRODUCTS OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS
42. " sin bu du :
a² + b²
(a sin bu – b cos bu) + C
43.
a" cos bu du =
a² + b²
(a cos bu + b sin bu) + C
POWERS OF u MULTIPLYING OR DIVIDING BASIC FUNCTIONS
51. ue" du = e" (u – 1) + C
Swe du =we
53. fu'a" du =
44. u sin u du = sin u – u cos u +C
Suco
u cos u du = cos u +u sin u + C
45.
52.
u"a"
|u° sin u du = 2u sin u + (2 – u²) cos u + C
u"-'a" du + C
In a
46.
In a
47.
u cos u du = 2u cos u + (u² – 2) sin u +C
e du
e du
54.
(n – 1)u“-1
n -
a" du
In a
a" du
* / u' cos u du
a"
is
n-1
48.
u" sin u du = -u" cos u + n
55.
(n – 1)un-I
du
In |In u| +C
u"
u" cos u du = u" sin u – n | u"-1 sin u du
56.u In u
49.
50. fu" lau
«" In u du
(n + 1)z [(n + 1) In u – 1] + C
POLYNOMIALS MULTIPLYING BASIC FUNCTIONS
57.
du
P'(u)e
[signs alternate: + - + -..]
|p(u) sin au du = -- p(u) cos au +p'(u) sinau +
| P(u) cos au du = - p(u) sin au + p'(u) cos au -
P" (u) cos au
[signs alternate in pairs after first term: ++-- ++--...]
58.
P"(u) sin au –
[signs alternate in pairs: ++ -- ++--..]
59.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce46589-331d-400c-a34f-bc9456c96fda%2Fd862d3f9-b93f-4b39-96f8-16665c16684c%2Fwcnfoxg_processed.png&w=3840&q=75)
