MAT 152
QUIZ 4
Compute the following integrals:
1. (3 points) This problem is done by integration by parts, where we integrate x and
differntiate arctan x:
Z
Z 2
x2
x
1
arctan x −
· 2
dx
2
2 x +1
Z
x2
1
x2
arctan x −
dx
2
2
x2 + 1
Z
x2
1
1
arctan x −
dx
1− 2
2
2
x +1
x2
1
arctan x − (x − arctan x) + C
2
2
Z
x arctan x dx =
=
=
=
2. (4 points) This is a regular trigonmetric integral problem. It can be done by writing
cos2 (x) = 1 − sin2 (x) and a substitution u = sin(x):
Z
4
Z
5
sin x cos x dx =
Z
=
Z
=
Z
=
sin4 x cos2 x
2
cos x dx
sin4 x 1 − sin2 x
2
cos x dx
u4 (1 − u2 )2 du
u4 − 2u6 + u8 du
u5 2 7 u9
− u +
+C
5
7
9
1
2
1
= sin5 x − sin7 x + sin9 x + C
5
7
9
=
3. (3 points)
Z
1
√
1 − x2 dx
−1
√
This problem requires no computation. Notice that the plot of y = 1 − x2 for x =
−1..1 is a semicircle. Indeed, squaring both sides we get y 2 = 1 − x2 or x2 + y 2 = 1,
equation for a circle of radius 1. See Figure 1.
The integral is half the area of the circle of radius 1 and equals π/2.
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Figure 1: Plot of y =
√
1 − x2 .
Note: this problem can be done by trigonometric substitution with x = sin u. This is
how you would get the indefinite integral, or definite integral with arbitrary bounds.
It was not required in this case if you understand what the integration does.
The idea of this problem is to avoid computation when you know what you are dealing
with.
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