section 5.7 inverse trigonometric functions: integration
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Section 5.7 Inverse Trigonometric Functions: Integration
Section 5.7 Inverse Trigonometric Functions:
Integration The most difficult step for recognizing the
integration of the inverse trig functions is to manipulate the functions so that they fit the form of the derivatives. This involves completing the square, a skill that we will have to recall from our Algebra days. Let’s look at an example.
Section 5.7 Inverse Trigonometric Functions:
Integration
0 0 0 0
22 22
2 2 2 2
0
0
22
2
2 2 2 214 8 4 4 4 2 4 4 2 14
112 arctan 2 arctan1 arctan 0 02 4 41
2 12
dx dx dx dx
x x x x x x
dxx
x
Section 5.7 Inverse Trigonometric Functions:
Integration We also have to recognize when the
derivative pattern is in front of us:
11 1 22 2
2 21 14 4 1
4
3 3 2 3arcsin 2
2 21 4 1 2
3 1 3 3arcsin1 arcsin
2 2 2 2 6 4 4 2
dx dxx
x x
Section 5.7 Inverse Trigonometric Functions:
Integration Now, some examples for you to try. I’ll be
quiet for awhile, please feel free to consult your neighbors.
3
23
2
20
3
2 1
1
9
cos
1 sin
dxx x
dxx
xdxx
Section 5.7 Inverse Trigonometric Functions:
Integration One last example here and this one is
kind of tough. I have it worked out on the next slide.
2
1
2
xdx
x x
Section 5.7 Inverse Trigonometric Functions:
Integration
2 2
2
1 12 2
2
2
2
1 1 2 2
22 2
2 2 2 2 2
1 2 2 1 1
2 2 221
22
x xdx dx
x x x xdu
u x x x du x dxdx
x dudx u du u C
ux xx
dx x x Cx x